THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


"his  b  -he 


iOUTHERN  BRANCH, 

UNIVERSITY  OF  CALIFORNIA, 

LIBRARY, 

LOS  ANGELES,  GALIF, 


m 


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THE 


DIRECTIONAL   CALCULUS, 


BASED    UPON    THE    METHODS    OF 


HERMANN    GRASSMANN. 


BY 


E.   W.   HYDE, 


PROFE880R  OP  MATHEMATICS    IN   THE   UnIVBRSITT   OF  CINCINNATI, 

Cincinnati,  Ohio. 


48607 

BOSTON,    U.S.A.: 

PUBLISHED  BY   GINN   &  COMPANY. 

1890. 


Copyright,  1890, 
By  E.  W.  HYDE. 


All  Rights  Reserved. 


Typography  by  J.  8.  Gushing  &  Co.,  Boston,  U.S.A. 


Pbesswork  by  Ginn  &  Co.,  Boston,  U.S.A. 


Engincerine  & 

Mathematical 
Sciences 
Library 


PREFACE. 


ri  iHE  wonderful  and  comprehensive  system  of  Multiple 
\  -'-  Algebra  invented  by  Hermann  Grassmann,  and  called 
^ty  by  him  the  Ausdehnungslehre  or  Theory  of  Extension,  though 
long  neglected  by  the  mathematicians  even  of  Grermany,  is  at 
the  present  time  coming  to  be  more  and  more  appreciated  and 
studied.  In  order  that  this  system,  with  its  intrinsic  natural- 
ness, and  adaptability  to  all  the  purposes  of  Geometry  and 
Mechanics,  should  be  generally  introduced  to  the  knowledge 
of  the  coming  generation  of  English-speaking  mathematicians, 
it  is  very  necessary  that  a  text-book  should  be  provided, 
suitable  for  use  in  colleges  and  universities,  through  which 
students  may  become  acquainted  with  the  principles  of  the 
^     subject  and  its  applications. 

The  following  pages  present,  in  part,  the  results  of  eight  or 
nine  years  of  study  and  experience  in  lecturing  to  university 
classes  upon  this  subject,  and  the  Author  hopes  that  they  may 
aid  in  some  measure  to  bring  about  that  general  study  and  use 
of  directional  methods,  which,  in  his  opinion,  will  ultimately 
cause  the  comparatively  awkward  and  roundabout  methods  of 
Cartesian  coordinates  to  be  superseded  by  them,  for  many  of 
the  purposes  of  analysis. 

As  the  great  generality  of  Grassmann's  processes  —  all 
results  being  obtained  for  ?i-dimensional  space  —  has  been 
one  of  the  main  hindrances  to  the  general  cultivation  of  his 


1 


iv  PREFACE. 

system,  it  has  been  thought  best  to  restrict  the  discussion  to 
space  of  two  and  three  dimensions. 

The  Author,  though  formerly  an  enthusiastic  admirer  of 
Hamilton's  Quaternions,  has  been  brought,  by  study  and  expe- 
rience in  teaching  both,  to  a  firm  belief  in  the  great  practical, 
as  well  as  theoretical,  superiority  of  Grassmann's  system. 
This  superiority  consists,  according  to  the  judgment  of  the 
writer,  first,  and  largely,  in  the  fact  that  Grassmann's  system 
is  founded  upon,  and  absolutely  consistent  with,  the  idea  of 
geometric  dimensions.  Second,  in  the  fact  that  all  geometric 
quantities  appear  as  independent  units,  viz. :  the  point ;  the 
point  at  00 ,  or  line  direction ;  the  definite  line  ;  the  line  at  oo , 
or  plane  direction ;  the  definite  plane ;  and,  finally,  the  plane 
at  00,  equivalent  to  a  volume,  which  is  scalar.  The  same 
holds  for  space  of  any  number  of  dimensions  ;  in  fact,  it  seems 
scarcely  possible  that  any  method  can  ever  be  devised,  compar- 
able with  this,  for  investigating  n-dimensional  space. 

Now  Quaternions  deals  only  with  the  vector,  or  line  direc- 
tion, and  the  scalar  —  for  a  quaternion  is  only  the  sum  of 
these  two;  it  knows  nothing  of  a  vector  having  a  definite 
position,  which  is  the  complete  representative  of  the  space 
qualities  of  a  force.  Further,  Hamilton's  vector  is  not  a  vec- 
tor pure  and  simple,  but  a  versor-vector  —  a  fact  which  gives  its 
peculiar  character  to  his  system,  as  being  really  a  calculus  of 
directed  imaginaries.  This,  which  Tait  regards  as  "  one  of  the 
main  elements  of  the  singular  simplicity  of  the  Quaternion 
Calculus,"  appears  to  the  writer  in  a  very  different  light.     It 

gives  rise  to  such  equations  as  -  =  Ji,  i.e.  multiplying  two 

J 
quantities  togetner  is  the  same  as  dividing  one  by  the  other, 

which  the  author  has  found  a  great  stumbling-block  to  the 

student.     It  can  hardly  be  regarded  as  a  natural  geometrical 


rr.EFACE.  V 

conception,  that  the  product  of  two  vectors  shoukl  be  another 
vector  perpendicular  to  each  of  them ;  still  less,  that  it  should 
be,  as  in  the  general  case,  a  scalar  plus  such  a  vector.  In  fact, 
the  Quaternion  system  practically  throws  overboard  the  idea 
of  geometric  dimensions. 

It  has  been  deemed  advantageous,  however,  to  make  use  of 
certain  terms  and  symbols  introduced  by  Hamilton,  such  as 
scalar,  tensor,  with  its  symbol  T,  etc. 

Although  this  work  is  based  upon  the  principles  and  meth- 
ods of  Grassmann,  yet  much  matter  will  be  found  in  it  that  is 
believed  to  be  original  with  the  author. 

Thus  the  idea  of  the  complement  in  a  point  system,  with  its 
geometric  interpretation,  as  developed  in  Arts.  40-44  and  63, 
does  not  occur  in  Grassmann's  works,  either  of  1844  or  1862. 
This  idea  is  of  great  value  in  geometric  applications,  and  cor- 
responds to  that  of  duality  in  Modern  Geometry. 

In  Arts.  68  and  69  are  treated  some  properties  of  what  I 
have  called  screivs,  which,  as  such,  are  not  discussed  by  Grass- 
mann. 

One  of  the  most  beautiful  and  valuable  theories  developed 
by  Hamilton  in  his  great  works  is  that  of  linear  and  vector 
functions.  This  theory  I  have  found  to  be  equally  capable  of 
application  to  point  functions  in  n-dimensional  space.  In 
Chapters  IV.  and  VI.  this  linear,  point  function  has  been  used 
in  the  discussion  of  the  conic  and  quadric,  giving  rise  to  a 
treatment  of  these  loci  bearing  the  same  analogy  to  trilinear 
and  quadriplanar  methods  that  the  vector  treatment  of  these 
curves  and  surfaces  bears  to  ordinary  Cartesian  methods,  and 
possessing  likewise  the  same  superiority  in  clearness,  concise- 
ness, and  intelligibility  which  the  vector  methods  have  over 
the  Cartesian. 


VI  PREFACE. 

It  appears  to  the  writer  that  Hamilton's  method  of  dealing 
with  linear  functions  much  excels  that  of  Grassmann,  as  devel- 
oped in  the  second  part  of  his  work  of  1862,  for  practical  util- 
ity and  convenience. 

It  is  hoped  and  believed  that  the  exposition  of  the  funda- 
mental ideas  and  principles  of  the  system  in  the  first  two 
chapters  will  be  found  sufficiently  full  and  explicit  to  give  the 
student  a  working  knowledge  and  grasp  of  the  subject,  such  as 
will  enable  him  to  take  up  without  difficulty  the  work  of  the 
succeeding  chapters,  or  to  read,  with  comparative  ease,  the 
original  treatises  of  Grassmann.  A  large  number  of  exercises 
have  been  inserted,  in  the  belief  that  only  by  the  repeated 
application  of  the  principles  which  he  has  learned  to  the  solu- 
tion of  actual  problems  can  the  student  acquire  any  real  com- 
mand of  any  branch  of  Mathematics.  Eight  or  nine  blank 
pages  will  be  found  at  the  end  of  each  chapter,  for  the  recep- 
tion of  notes,  solutions,  etc.,  the  Author's  experience  having 
convinced  him  of  the  advantages  of  such  an  arrangement. 

E.   W.  HYDE. 


TABLE   OF   CONTENTS. 


CHAPTER   I. 

ADDITIOX    AND    SUBTRACTION. 

PAGE 

Definition  of  vector,  plane- vector,  etc 1 

Equality 1 

The  point,  equality  of  points 2 

Difference  of  unit  points  2 

Sum  and  difference  of  vectors 3  ■ 

Addition  of  points 4 

Case  in  which  2m  =  0 5 

Conditions  that  two  vectors  may  be  parallel,  and  that  three  may  be 

parallel  to  one  plane 7 

Dependence  and  independence 8 

Tensor,  unit,  reference  systems 9 

Exercises 10 


CHAPTER   II. 

MULTIPLICATION. 

Fundamental  notion  of  a  geometric  product 23 

Posited  quantities  23 

Definitions  of  combinatory  products 24 

Laws  of  combinatory  multiplication 25 

Product  of  two  points 25 

Product  of  two  vectors 20 

Product  of  three  points 27 

Product  of  three  vectors 28 

Product  of  four,  or  more,  points,  or  vectors 29 

Equations  of  condition,  planimetric  products 30 

Product  of  points  expressed  in  terms  of  reference  points 31 

Planimetric  product  of  two  lines 32 

Product  of  a  point  and  two  lines 33 


Viii  TABLE   OF   CONTENTS. 

FAOK 

Conditions  L^L^  —  0  and  L^L^L^  =  0 34 

Product  of  two  II  lines  and  of  two  vectors 34 

Sum  of  point- vectors 35 

Complement 36 

Co- product  and  co-square 37 

Complement  in  a  plane  point  system 39 

Geometric  interpretation  of  the  same 40 

Distance  between  two  points 41 

Anti-polar  property  of  complement 41 

Projections 43 

Multiplication  table  for  a  point  system 44 

Division  indeterminate 49 

Exercises 50 

Stereometric  products 56 

Products  of  points  expressed  in  terms  of  reference  points 56 

Product  of  a  line  and  a  plane 57 

Product  of  two  planes;  of  tliree  planes 58 

Product  of  four  planes;  products  of  plane-vectors 59 

Equations  of  condition 60 

Addition  of  planes  and  plane- vectors 60 

Addition  of  point-vectors,  or  lines 62 

The  complement  in  three-dimensional  space 62 

Complement  in  a  point  system  in  three-dimensional  space 64 

Condition  that  a  linear  function  of  the  edges  of  the  reference  tetrac- 

dron  shall  be  a  point-vector 65 

Point  system  in  solid  space  a  fifteen- fold  algebra 66 

Projections  in  a  vector  system 67 

Projections  in  a  point  system 68 

Normal  form  of  the  screw 71 

Product  of  two  screws 72 

Products  Spi  •  Spj  and  Spi  •  Sp.^  •  Sp^ 73 

Quaternion  expressions  corresponding  to  some  expressions  of  this 

calculus  73 

Exercises 74 


CHAPTER   III. 

APPLICATIONS    TO    PLANE    GEOMETRY. 

Preliminary  statements 87 

Equations  of  plane  space 88 

Equations  of  right  lines 89 


TABLE   OF  CONTENTS.  IX 

PAOK 

Transformation  from  a  point  to  a  vector  system,  and  vice  versa 90 

Exercises 91 

Equations  L^p  •  L.j,p  =  C  and  p^L  •  p^L  =  C 92 

Conies  tlirougii  tliree  points,  and  touching  tliree  lines 93 

Differentiation  94 

Examples  of  differentiation 96 

Tangent  and  normal.     The  circle 97 

Exercises  on  the  circle 98 

The  parabola 99 

Exercises  on  the  parabola 101 

The  ellipse  and  hyperbola 102 

Tangent  and  normal.     Diameter 104 

The  <p  function 105 

Exercises  on  the  (p  function 107 

The  ^^  function 108 

Poles  and  polars 109 

Exercises 110 

The  conic  referred  to  conjugate  diameters Ill 

Area  of  circumscribed  parallelogram  whose  sides  are  conjugate  in 

direction 112 

Exercises  on  the  central  conies 112 

The  general  linear,  vector  function  in  plane  space 113 

Inversion  of  the  <(>  function 114 

The  invariants  mg  and  m^ 116 

Exercises  on  inversion 117 

The  general  equation  of  the  second  degree  in  plane  space 117 

Center  of  the  locus 118 

Case  when  <p\(pfi  =  0  119 

Axes  of  the  locus 120 

Exercises  on  the  general  conic 122 


CHAPTER   IV. 

SCALAK    POINT    EQUATIONS    OF    THE    SECOND    DEGREE    IN    PLANE    SPACE. 

All  equations  homogeneous 133 

Order  of  the  curve  p\<pp  =  0 133 

Diameters  134 

Tangent  and  polar 136 

Center.     Conjugate  points 136 

Normal  system  of  conjugate  points  . .  ..^^.^.^ 137 


X  TABLE   OF   CONTENTS. 

PAGE 

Canonical  form  of  ^|^ 139 

Condition  that  p\<pp  shall  be  the  product  of  two  linear  factors 140 

Nature  of  the  locus  at  oo  140 

Most  general  form  oip\<pp 141 

Conic  through  the  reference  points 141 

Circle  through  the  reference  points 142 

Exercises 14;] 

Equations  p\(pp  =  C  and  p\<i>p  =  0 , 144 

Anti-polar  of  any  point.     Reciprocating  ellipse 14j 

Complement  of  any  point 146 

Line  equations 147 

Center  of  the  curve  Z|i^Z=  0 147 

Nature  of  the  curve  L\\l/L  =  0 148 

Pascal's  and  Brianchon's  theorems 149 

Inversion  of  ^ 150 

Exercises 151 


CHAPTER  V. 

SOLID    OEOMETRV. 

Preliminary  statements 163 

Non-scalar  point  equations  of  surfaces  and  curves  in  general 164 

Non-scalar  plane  equations  of  surfaces  in  general 164 

Non-scalar  vector  equations  in  general 165 

Equations  of  planes,  hues  and  points 165 

Vector  equations  of  planes  and  lines 166 

Exercises 167 

The  sphere 170 

Exercises  on  the  sphere 171 

The  paraboloids 172 

Rectilinear  generators  173 

Exercises  on  the  paraboloids 173 

The  central  quadrics 174 

Tangent  plane  and  normal 176 

Diametral  plane  and  conjugate  diameters 177 

Poles  and  polars 178 

Volume  of  circumscribed  parallelopiped 179 

Surface  refeiTed  to  conjugate  diameters 180 

Rectilinear  generators 180 

Exercises 181 

Condition  that  p\<pp  shall  be  factorable 183 


TABLE   OF  CONTENTS.  XI 

PAGE 

The  <p  function  in  general.     Inversion 183 

The  general  scalar  equation  of  the  second  degree  in  terms  of  vectors.  184 

Cyclic  sections  185 

Exercises 186 

Center  of  the  general  quadric  surface 188 

Maximum  and  minimum  values  of  Tp 190 

Roots  of  the  discriminating  cubic 192 

The  discriminant  WqC' 194 

Table  of  classification  of  the  quadric 195 

Exercises 195 


CHAPTER    VI. 

SCALAR   POINT   EQUATIONS    OF    THE    SECOND    DEGREE    IN    SOLID    SPACE. 

Differentiation.     The  general  homogeneous  equation 207 

Diametral  planes.     Significance  of  the  quantity  \(pp 208 

Center  of  the  surface 209 

Sets  of  conjugate  points 209 

Normal  set  of  conjugate  points 210 

Solution  of  the  equation  <pp=  np 210 

Canonical  form  oi  p\^p 212 

Rectilinear  generators 212 

The  discriminant 213 

Nature  of  the  surface  at  infinity 214 

Table  of  classification 216 

Quadric  through  the  four  reference  points   217 

Conditions  that  (496)  shall  represent  a  sphere 217 

Exercises 218 

Inversion  of  <^ 222 

Equation  of  anti-polar  plane.     Reciprocating  ellipsoid 223 

Proof  that  \p  is  the  anti-polar  plane  oip 224 

Scalar  plane  equations 225 

Center  of  surface  P|;|.P=  0 225 

Reciprocal  surfaces.     The  discriminant  as  a  criterion 226 

Table  for  the  determination  of  P\^P  =  0 227 


xii  TABLE  OF   CONTENTS. 


CHAPTER   VII. 

APPLICATIONS   TO    STATICS. 

PAGX 

Introductory  statements 237 

Forces  acting  on  a  particle 238 

Equilibrium  of  particle  on  a  smooth  curve 238 

Equilibrium  of  particle  on  a  smooth  surface 239 

Examples 240 

Forces  acting  on  a  rigid  body 242 

Parallel  forces.     Couples 242 

Conditions  for  a  single  resultant  and  for  equilibrium 243 

Normal  fonn  of  a  wrench .*.  243 

Reduction  of  a  wrench  to  the  sum  of  two  forces 245 

Exercises 245 


DIRECTIONAL  CALCULUS. 


CHAPTER   I. 


ADDITION   AND   SUBTRACTION. 

1.  The  quantities  to  be  treated  in  this  book  are  such  as 
possess  one  or  more  of  the  qualities  magnitude,  direction, 
position. 

A  quantity  possessing  magnitude  only,  though  it  may  be 
either  positive  or  negative,  is  a  scalar  quantity. 

A  quantity  possessing  magnitude  and  direction  is  a  directed, 
or  vector,  quantity.  Thus  a  line  of  given  length  and  direction 
is  a  line-vector,  or  simply  a  vector;  a  plane  area  of  given  direc- 
tion and  magnitude  is  a  plane-vector. 

If  a  vector  pass  through  some  definite  point,  it  has  position, 
as  well  as  magnitude  and  direction,  and  may  be  called  a  x>oint- 
vector.  Similarly,  when  a  plane-vector  passes  through  a  defi- 
nite point,  it  may  be  called  a  point-pla-ne-vector.  We  shall, 
however,  often  substitute  for  point-vector  and  point-plane- 
vector  the  simpler  terms  line  and  plane,  especially  when,  as  is 
often  the  case,  the  question  of  magnitude  does  not  concern 
us.  These  terms  correspond  respectively  to  the  terms  Linien- 
theil  and  Flachentheil,  as  used  by  Grassmann. 

2.  Equality.  Two  quantities  are  said  to  be  equal  when  their 
qualities  are  identical.  Thus  two  vectors  having  the  same 
length  and  direction  are  equal.  Of  course  the  term  direction 
must  include  the  idea  of  sense^ — 


2  DIRECTIONAL  CALCULUS.  [Art.  3, 

3.  The  point.  A  point,  as  used  in  this  Calculus,  may  be 
defined  as  a  position  in  space,  of  no  magnitude,  but  endowed 
with  a  certain  value  which  will  be  called  its  weight.  This 
value,  or  weight,  is  a  scalar  quantity,  and,  as  such,  obeys,  in 
connection  with  the  point  to  which  it  belongs,  the  ordinary 
laws  of  multiplication ;  that  is,  if  j>  be  u  point  and  m  and  n 
weights,  mp  =  pm  and  mp  +  np  =  (m  -f  t^)p-  Letters,  such  as 
p,  p^  etc.,  will  be  used  to  designate  the  positions  of  points,  and 
these  will  be  multiplied  by  other  letters,  such  as  k,  I,  m,  n, 
representing  their  weights.  If  the  weight  of  a  point  be 
unity,  the  figure  1  representing  this  weight  will  be  omitted,  so 
that  a  letter  without  coefficient,  representing  a  point,  will  be 
regarded  as  representing  a  point  of  unit  weight,  or  a  unit 
point. 

4.  Equality  of  jyoints.  If  two  weighted  points  rtiiPi  and 
m2P2  are  to  be  equal,  their  qualities  must  be  identical.  These 
are  their  weights  ((/ttasi'-magnitudes)  and  their  positions. 
Hence,  in  order  that  viiPi  and  m2P2  may  be  equal,  we  must 
have  Ml  =  TJijj  and  2>,  coincident  Avith  p^. 

5.  Difference  of  unit  points.  Suppose  wii  =  ma  =  1,  but  pi 
and  p2  not  coincident.  Then  the  only  difference  between  p^  and 
P2  is  one  of  position.  This  difference  is  naturally  expressed  by 
saying  that  it  is  a  certain  distance  in  a  certain  direction,  which 
corresponds  precisely  with  the  definition  of  a  vector  in  Art.  1. 
Thus  a  difference  of  position,  which  is  the  only  difference 
between  two  unit  points,  is  naturally  represented  by  a  vector. 
Now  we  may  extend  the  meaning  of  the  algebraic  sign  for 
a  difference,  i.e.  the  minus  sign,  so  that  it  shall  indicate  the 
difference  of  position  of  two  unit  points,  provided  that  this 
use  of  the  sign  leads  to  no  inconsistencies  or  contradictions. 
Hence,  if  pi  and  2>2  are  unit  points,  and  c  the  vector  from  jh  to 
P2,  we  write 

i>2-i>i  =  c, (1) 

and  the  development  of  the  subject  will  show  this  to  be  in 
accordance  with  the  above  proviso. 


Chap.  I.] 


ADDITION   AND   SUBTRACTION. 


If  2h  —Pi  =  £  =  0,  or  P2  =l'>\i  there  is  no  diiference  of  position 
between  the  points  ;   i.e.  they  coincide,  as  previously  stated. 
Adding  j>i  to  both  sides  of  (1),  we  have 


lh  =  lh  +  ^, 


(-0 


in  which,  of  course,  the  meaning  of  the  sign  +  is  extended  in 
a  manner  similar  to  that  of  the  minus  sign. 

Eq.  (2)  shows  that  the  sura  of  a  point  and  a  vector  is  a  point, 
distant  from  the  first  point  by  the  length  of  the  vector,  and  in 
the  direction  of  the  same. 


6.  Let  e„  62,  63  be  any  three  unit  points,  and  ci,  c,,  eg  vectors 
II  and  equal  in  length  to  the  sides  of  the  triangle  ^162^3  taken  in 
order,  so  that  cj  =  63  —  e^,  £2  =  Cj  —  63,  £3  =  62  —  Ci  •  then 


Ej  +  £,  +  £3  =  63  —  62  +  61  —  Cg  +  62  —  61  =  0 ; 


{^) 


that  is,  the  sum  of  the  vector  sides  of  a  triangle,  taken  posi- 
tively in  the  same  sense  around  the  triangle,  is  zero. 
We  have  also 

£1  -f  £2  =  ^1  —  62  =  —  £3 ; (4) 

so  that  the  sum  of  two  vectors  is  a 
vector  which  represents  the  differ- 
ence of  the  initial  and  final  posi- 
tions of  a  point  which  moves  along 
the  two  vectors  successively. 

Write       «/  =  62  —  ^3  =  —  £1 ; 

then  £1'  — £2=^2— 63  — 61  +  63  =  62  — 61  =  £3;    .     .     .(4  a) 

that  is,  the  difference  of  two  vectors  is  the  vector  joining  their 
extremities  when  they  are  drawn  outwards  from  a  common 
point. 

Similarly,  if  we  take  n  unit  points  gj,  e^,  •••  e„,  we  have 

(62  -  61)  +  (^3  -  62)  +  (^4  -  ^3)  +  •  •  •  +  (e„  -  e„-i)  =  6„  -  61 ; 
that  is,  the  sum  of  any  nvimbev  of  vectors  is  the  vector  repre- 


4  DIRECTIONAL  CALCULUS.  [Akt.  7. 

senting  the  difference  between  the  initial  and  final  positions  of 
a  point  which  moves  along  all  the  vectors  in  succession.  The 
result  is  evidently  independent  of  the  order  in  which  the 
vectors  are  taken. 

7.  Addition  of  points.  In  order  to  interpret  the  meaning 
of  such  an  expression  as  niiPi  +  m^jh  +  etc.  =  Swip,  we  will 
assume,  — 

1st.  That  the  sum  of  any  number  of  points  will  be  itself  a 
point,  which  we  shall  call  the  mean  point  of  the  system. 

2d.  That  the  weight  of  the  sum  will  be  equal  to  the  algebraic 
sum  of  the  weights  of  the  points. 

These  assumptions  are  allowable  if  they  lead  to  no  contra- 
dictory or  inconsistent  results.  In  accordance  with  them  we 
write  for  two  points, 

miPi  +  m2P2  —  (mi  +  m2)p 

Transposing,  we  have 

in2{P2-p)  =  '>MP-Pi), (^>) 

which  shows  that  p  is  on  the  straight  line 

joining  pi  and  p^,  at  distances  from   these    i ' 

points  inversely  proportional  to  their  weights.     '  ^ 

Let  e  be  any  point  whatever,  and  subtract  from  both  sides  of 
(5)   (mi  +  m2)e; 

•••  wii(pi  —  e)  -f  m2(p2  —  e)  =  (Wi  +  ?)i2)  (p  -  e), 
or  p-e=      ^^'      {p,-e)-i-      ^^      (p2-e),  .     .     (7) 

by  which  p  is  easily  found. 
For  three  points  we  have 

miPi  +  m2P2  +  ms2h  =  (mi  +  fn2  +  ms)(p)',      .     .     (8) 

or,  by  (5), 

(mj  -f  m2)p  -f  rUsPs  =  (mj  +  m,  +  mj)  (p) ' ; 


CuAF.  I.]  ADDITION   AND   SUBTRACTION.  5 

SO  that  (p) '  is  on  the  line  joining  p  and  p^  at  distances  from 
them  inversely  proportional  to  their  weights. 

Subtracting   from   both   sides   of  (8)   (mi  +  m2  +  m3)e,   we 
have,  on  dividing  by  %m, 

by  which  (p) '  is  easily  constructed. 

Similarly,  for  any  number  of  points  we  have 

2mj3=j32m, (10) 

and  »  — e  =  — — 2[m(2)  — e)] (11) 

The  reader  will  notice  the  analogy  to  "center  of  parallel 
forces." 

8.    Case  in  which  5m  =  0.     In  (5)  and  (6)  let 

)  wij  +  m^  =  0, 

id  these  equations  become 

wi2(i>2-i5i)  =  0-p       (12) 

^ud  p  —  P2=P—I>i •     •   (13) 

If  wij  be  not  zero,  and  x>\  not  coincident  with  p,2,  these  equations 
can  only  be  satisfied  when  p  is  at  an  infinite  distance  on  the 
line  piP2-  Eq.  (6)  shows  at  once  that,  when  m^  is  negative, 
p  is  not  between  pi  and  p2,  and  that,  as  the  numerical  value  of 
wig  approaches  that  of  mj,  p  recedes  farther  and  farther,  until, 
when  mi  +  m^  =  0,  p  is  at  go.  Thus  the  meaning  of  eq.  (12) 
is,  that  a  point  of  zero  weight  at  an  infinite  distance  is  equivalent 
to  a  vector  of  definite  length  directed  towards  this  point.  It  will 
often  be  convenient  to  regard  vectors  as  points  at  go. 

Consider  next  the  general  case  of  n  points.     Eqs.  (10)  and 
(11)  become,  when  :Si  m  =  0, 

2imp  =  0-j3 


and  Si  [m(p  — e)]  =0  •  (p  — e). 


DIRECTIONAL  CALCULUS.  [Art.  8. 


Let 
then,  since 


(15) 


^2  wip  =  (p) '  2" m  =  ?M'(i>) ',  say  : 

mi  +  m'  =  2"?Ji  =  0, 

2" mp  =  rriipi  +  m'( j?)'  =  m'  ((p)'  — i^i)  =  0  -p,  ^ 

and       2r[m(p  — e)]  =??ii(pi  — e)  +  m'((J9)'  — e) 
=  m'({py-p,)  =  0-{p-e) 

Hence  the  direction  of  the  point  at  oo  is  found  by  constructing 
the  mean  point  of  all  the  points  except  one,  when  the  vector 
from  the  excepted  point  to  this  mean  point  has  the  required 
direction.  As  j>  is  a  unique  point  for  any  given  system  of 
weighted  points,  and  2h  ^^J  be  taken  as  any  point  of  the 
system,  it  follows  that,  in  any  system  of  points  whose  total 
weight  is  zero,  a  line  drawn  from  any  point  of  the  system  to 
the  mean  point  of  all  the  rest  is  parallel  to  any  other  such 
line. 

Finally,  suppose       2mp=p'2m  =  0, (16) 

which  gives  as  a  necessary  consequence  2m  =  0.  Then,  by 
(15),  if  m'  is  not  zero,  we  must  have 

(py=Pi; 

that  is,  pi  is  the  mean  point  of  the  system  consisting  of  the 
remaining  points.  But,  as  jh  ^^J  be  anr/  point  of  the  system, 
it  appears  that  any  one  of  the  points  is,  in  this  case,  the  mean 
of  all  the  rest.  For  example,  let  w  =  3 ;  then  (16)  becomes 
niiPi  -\-  rriiPi  +  mgpg  =  0,  which  requires  also  mi  +  mg  +  wig  =  0; 
whence 

rriiPi  +  m2P2  =  -  nisPs  =  (wij  +  «i2)i>!» 

or  ps  is  the  mean  point  of  miPi  and  rriiPi-  Now  we  have  seen 
in  Art.  7  that  the  mean  of  two  points  is  collinear  with  them ; 
hence 

'miPi  +  m2P2  +  insP3=0 (17) 

is  the  condition  thai  the  three  points  shall  be  collinear. 


Chap.  L]  ADDITION   AND   SUBTRACTION.  7 

If  three  points  pi,  p^,  Pz  are  not  collinear,  and  yet  are  con- 
nected by  a  linear  relation  such  as  (17),  this  equation  can  only 
be  satisfied  when  each  iveight  is  separately  zero,  that  is 

7Jii  =  m2  =  W3  =  0 ; (17  a) 

for,  as  we  have  just  seen,  for  all  values  of  the  weights  different 
from  zero  the  three  points  are  coUinear,  but  when  each  weight 
is  zero,  (17)  is  satisfied  independently  of  the  positions  of  the 
points.  • 

Similarly,       wiii?!  +  mzPg  +  w^sPs  +  wi4P4  =  0,      .     .     .   (18) 

which  requires  also  wii  +  m2  +  wig  +  mi  =  0,  is  the  condition 
that  four  points  shall  be  coplanar,  and  if  the  points  are  not 
in  one  plane,  then  we  must  have 

7ni  =  nio  =  WI3  =  mi  =  0 (19) 

A  similar  condition  between  Jive  points  would  imply  that 
the  points  were  all  in  one  tri-dimensional  space,  which  leads  to 
the  consideration  of  space  of  higher  dimensions  than  three. 

9.   Let  €1  and  €2  be  two  vectors,  and  ni  and  ^2  scalars ;  then 

nici  +  w2£2  =  0 (20) 

is  the  condition  that  the  vectors  shall  be  parallel,  as  appears  at 
once  by  Art.  2,  if  we  write  the  equation  ji^ci  =  —  W2C2. 

Similarly,        ?iiei  +  jij^a  +  ^^s  =  0 (21) 

requires  the  three  vectors  to  be  parallel  to  one  plane;  for,  by 
Art.  6,  the  vectors  ^ijq,  ?i2f25  ^^s^s)  iiiay  be  represented  by  the 
three  sides  of  a  triangle.  The  same  appears  from  (17)  if  we 
take  pi,  P2,  Ps  as  points  at  ao. 

If  we  have  the  additional  condition 

^1+^2  + "3  =  0, (22) 

the  extremities  of  the  vectors,  if  drawn  outwards  from  a  point 
will  be  in  one  right  line.     For  let  eo,  Ci,  62,  e^  be  four  points 


8  DIRECTIONAL  CALCULUS.  [Akt.  10. 

SO  taken   that   gj  — e,)=£i,  Cj— 60=^2?  e^  — 60  =  63;   then   (21) 
becomes 

Wi(ei  —  e„)  +  Ho(e2  —  eo)  +  >i3(<?3  —  ^o)  =  W, 

or,  by  (22),         niCi  +  71262  +  Jia^s  =  0  ; 

which,  by  (17),  requires  ej,  62?  ^3  to  be  collinear. 

Similarly,  it  may  be  shown  that  the  eqs. 

»  njCi  +  ?l2e2  +  '*3C3  +  'M4  =  0 

111  +  7I2  +  Wg  +  ?J4 


+  7^4  =  0| 
J4  =  0  I 


(23) 


are  the  conditions  that  the  extremities  of  the  vectors  tj,  cg,  etc., 
drawn  outwards  from  one  point  shall  be  coplanar. 

10,  If  the  equation  ^mp  =  0  can  be  satisfied  by  finite  values 
of  the  m's  different  from  zero,  then,  as  we  have  seen,  any  one 
of  the  points  can  be  expressed  in  terms  of  the  others,  and  is, 
therefore,  dependent  on  them;  thus  the  points  are  mutually 
dependent.  If,  however,  the  equation  can  only  be  satisfied 
when  all  the  m's  are  zero,  no  relation  of  this  kind  exists  be- 
tween the  points  ;  i.e.  they  are  independent.  This  is  evidently 
as  true  of  vectors  as  of  points. 

Thus,  by  Art.  4,  two  different  points  are  always  independent. 

Three  non-coUinear  points  are  independent,  while  three  col- 
linear points  are  mutually  dependent. 

Four  non-coplanar  points  are  independent,  while  four  coplanar 
points  are  mutually  dependent. 

Any  five,  or  more,  points  are  always  mutually  dependent. 

Similarly,  two  non-parallel  vectors  are  independent,  but  two 
parallel  vectors  are  dependent. 

Three  vectors  not  parallel  to  one  plane  are  independent,  but 
when  all  are  parallel  to  one  plane  they  are  mutually  dependent. 

Four,  or  more,  vectors  are  always  mutually  dependent. 

In  a  system  of  independent  points  or  vectors  no  one  can  be 
expressed  linearly  in  terms  of  the  others. 


Chap.  T.]  ADDITION   AND   SUBTRACTION.  9 

11.  The  tensor  of  a  directed  quantity  is  its  numerical  magni- 
tude, taken  always  as  a  positive  quantity.  It  will  be  denoted, 
as  in  Quaternions,  by  T ;  as,  Te.  =  tensor  of  e. 

That  portion  of  a  directed  quantity  whose  magnitude  is 
unity  will  be  called  its  unit,  and  will  be  denoted  by  U;  as, 
C7e  =  unit  of  e. 

Hence  we  have 

T€-U€=€ .      (24) 

12.  Reference  systems.  Let  }}  be  any  point,  and  e^,  e^,  e^ 
three  fixed  reference  points ;  then  writing 

2-)  =  xeo  +  ye,  +  ze2,       (25) 

we  must  have,  if  p  is  to  be  a  unit  point, 

x  +  y  +  z=^l, (26) 

and  p  is  the  mean  point  of  xe^,  ye„  and  ze^.  Eliminating  x 
between  (25)  and  (26),  we  have 

P  =  eo  +  y{ei-e^)-\-z{e.2-e^),      ....     (27) 

from  which  it  appears  that,  by  varying  y  and  z,  p  may  be  made 
to  occupy  any  position  whatever  in  the  plane  of  e^,  Cj,  62- 
Hence  any  three  unit  points  e^,  Ci,  e^  may  be  taken  as  a  refer- 
ence system  for  plane  space,  in  terms  of  which  all  poiffts  of 
this  plane  space  may  be  expressed. 

Writing  p  —  eo  =  p,  e,  —  Cq  =  cj,  eg  —  Co  =  eg,  eq.  (27)  becomes 

p  =  i/ci  -f  Z€., (28) 

and  any  vector  in  plane  space  may  be  expressed  in  terms  of 
two  reference  vectors  c^,  e,  in  that  plane.  When  cj  and  cj  a^re 
of  unit  length  and  at  right  angles,  the  system  will  be  called  a 
unit  normal  reference  system,  and  ii  and  12  will  be  substituted 
in  this  case  for  e^  and  ej* 

Similarly,  in  solid  space  any  point  p  may  be  expressed  in 
terms  of  four  non-coplanar  points  e^,  gj,  62,  63,  by  the  equation 

p  =  ivco  +  xet,  -f  2/62  +  zes, (29) 


10  DIRECTIONAL   CALCULUS.  [Art.  13. 

which  requires,  if  2'  is  a  unit  point, 

w  +  x-\-y  +  z  =  l, (30) 

whence  P  =  eo  +  x{ei  — eo)  +  y(e2  — eo)-\- z^e^—e^).      (31) 

Thus  any  four  unit  j)oints  may  be  taken  as  a  reference  sys- 
tem for  solid  space. 

Putting,  as  before,  p=p  —  e^,  tj  =  gj  —  e^,  etc.,  (31)  becomes 

p  =  xe,+yu-\-zc„ (32) 

so  that  any  vector  may  be  expressed  in  terms  of  three  given 
vectors  e^,  £2,  £3.  When  Tcj  =  Tc^  =  ^^3  =  1,  and  the  three  vec- 
tors are  at  right  angles  to  each  other,  we  have  a  unit  normal 
system,  and,  in  this  case,  substitute  ij,  ic,,  13  for  tj,  cg,  cg. 

13.   A  number  of  exercises  will  now  be  given  illustrative  of 
the  application  of  the  preceding  principles. 

(1)  The  mean  point  of  unit  points  at  the  vertices  of  a  tri- 
angle  coincides   with   the   center  of 

gravity  of  its  area.     The  center  of  An^^^          e„+e, 

gravity  of  the  area  will  be  at   the  /  \  ^^^~    2 

common  point  of  lines  through  the  -~-i=p/- — Ix^:^ -~>e 

vertices   and  the    middle    points   of  /  /  j^^-^'e^ej 

the  opposite  sides.      Thus,  if  p  be  l/^-^^'^^'^'    ^ 

the  point,  ^ 

p=xe^-\-yp^=x'e.,+y'p^=xe^+  ^  (62+63)  =•'^'62+^  (63+61). 

••■   {x  -  ^y')e,  +  (12/  -  x')e,  +  i(y  -  y')e,  =  0  ; 
by  eqs.  (17)  and  (17  a)  this  gives 

X  -  W  =hj  -  ^'  =  y  -  y'  =  0. 
But,  since  p  is  a  unit  point,  we  have 

x  +  y  =  l  =  x'  +  y\ 
From  these  equations  we  find 

x  =  x'  =  ^,   y  =  y'  =  ^', 
whence  P  =  iei +  |p,  =  i((?i +  62  +  63), 


Chap.  L] 


ADDITION   AND    SUBTRACTION. 


11 


which  shows  that  p  is  the  mean  point  of  ei,  e^,  eg,  and  trisects 
the  distance  from  ei  to  pi. 

(2)    To  find  the  common  point  of  the  perpendiculars  from 
the  vertices  of  a  triangle  on  the  opposite  sides. 

Let  I,  m,  n  be  the  ratios  of  the 
sides  of  the  triangle  to  the  cosines 
of  the  opposite  angles  ;  i.e. 

T(e,-e,)  . 

cos(Z  at  ei) 


then 


Pi  = 


m  +  71 


n  +  l  '  I  -\-m   ' 

Proceeding  now  precisely  as  in  the  first  problem,  with  the 
above  values  of  ^^i,  p2,  2h  instead  of  those  there  used,  we  find 

,  _  Ze^  +  wifig  +  W6-! 
l  +  m  -\-  n 

(3)    Find  the  common  point  of  perpendiculars  to  the  sides 
of  a  triangle  through  the  middle  points  of  the  respective  sides. 
By  the  figure  for  the  last  problem  we  have 

P"  =  K«i  +  ^2)  +  ^(^3  -P')  =  ^(^2  +  63)+  yi^i 
l(e^  —  e,)  +  m(€s  —  e.2) 


■P') 


I  -\-m  +  n 

??i(pi  — 62)  + ?i  (61  —  63) 

l-\-in-\-  n 

Hence,  equating  to  zero  coefficients  of  gj  and  e,, 

1  _        xl        _y  (m  +  n)  _  /^  _  1 xm 

2  l  +  m  -\-n      I  +  m  +  n  2 


l  +  m-\-  n 

1  I      y^     , 

2  l-{-m  +  n 


.   x=v=^  and  u"  _(^  +  n)e,  +  (n  +  l)e,_  +  (l  +  m)e3. 
"  '      ^     2'  ^  2{l  +  m  +  n) 


12  DIEECTIOx,AL   CALCULUS.  [Art.  13. 

(4)  Show  that  p,  p\  p"  of  examples  1,  2,  3  are  collinear. 

It  is  easily  seen  that  3p—p'  —  2p"  =  0,  which,  by  (17), 
proves  the  collinearity  of  the  points.  It  is  evident  also  that 
p,  p',  and p"  are  collinear  whatever  values  be  assigned  to  l,m,n', 
for  nothing  in  the  demonstration  depends  on  /,  m,  n  having 
the  values  assigned  in  ex.  2. 

(5)  The  center  of  gravity  of  the  volume  of  a  tetraedron 
coincides  with 

(a)  The  mean  of  unit  points  at  the  vertices ; 

(b)  The  center  of  gravity  of  the  tetraedron  whose  vertices 
are  the  centers  of  gravity  of  its  faces ; 

(c)  The  mean  of  the  middle  points  of  its  edges. 

(G)  Find  a  point  such  that  the  sum  of  the  vectors  drawn 
from  it  to  n  given  points  shall  be  zero. 

Ans.  p  =  -Ste  =  mean  of  points. 
n 

(7)  Construct  by  eq.  (11)  the  mean  point  of  points  at  the 
four  corners  of  a  square,  whose  weights  are  respectively 
1,  2,  3,  and  4. 

(8)  Let  ei,  e^,  Cg,  64,  be  the  corners  of  a  parallelogram,  and 
65  =  ^(gj  4-  e,) :  show  that  ejCg  and  6465  will  trisect  each  other. 

(9)  Let  gj,  62,  ^3,  64  be  the  corners  of  a  parallelogram,  and  let 
p^,  P2  be  points  on  a  line  ||  to  ejCo :  if  Qi  is  the  common  point  of 
PiCi  and  p2e2,  and  (?2  the  common  point  of  pie^  and  pze^,  show 
that  QiQi  is  II  to  6164. 

(10)  Let  61,  62,  63,  64  be  four  non-coplanar  points,  and  let 
Pu  Pd  Psi  Pi  t)e  points  taken  on  6162,  6263,  6364,  and  6461  respec- 
tively: find  the  condition  to  be  fulfilled -in  order  that  2hP2} 
6163,  and  psPi  may  have  a  common  point. 

Write  pi  =  miBi  +  71,62,  P2  =  ^262  +  ^2^3?  etc. ;  then  the  p's 
must  be  coplanar,  which  leads  to  the  condition 


Chap.  I.]  ADDITION  AND   SUBTRACTION.  13 

(11)  If  through  any  point  within  the  triangle  Cie^eg  lines  be 
drawn  ||  to  the  sides  and  terminated  by  them,  and  if  I,  m,  n,  be 
the  respective  ratios  of  these  lines  to  the  sides  to  which  they 
are  || ;  then  l  +  m  +  n  =  2. 

(12)  The  center  of  gravity  of  the  sides  of  a  triangle  coin- 
cides with  the  center  of  the  circle  inscribed  in  the  triangle 
formed  by  joining  the  middle  points  of  the  sides. 

(13)  Find  the  center  of  gravity  of  the  faces  of  a  tetraedron ; 
also  of  the  edges. 


Chap.  II.]  MULTIPLICATION.  23 


CHAPTER   II. 

MULTIPLICATION. 

14.  Grassmann's  first  conception  of  a  geometrical  product  is 
that  it  is  what  is  jyi'oduced  or  generated  by  the  first  factor,  as 
it  moves  over  a  distance  determined  by  the  second. 

Thus  cjCo,  if  ci  and  cj  ^^'^  two  vectors,  signifies  the  directed 

plane  area  bounded  by  the  parallelogram  

whose  sides  are  ||  and  equal  in  length  to  /  / 

ci  and  Co ;  that  is,  the  plane  area  gener-         /^^  / 

ated  by  tj  as  it  moves,  ||  to  itself,  along     /     » / 

€2  from  its  initial  to  its   final   point.  ^' 

This  is  a  plane-vector  as  defined  in  Art.  1.  The  product  will 
evidently  have  the  same  value  whether  the  initial  point  of  ci 
moves  in  the  direction  e^  or  ^^  ^^J  other  path,  provided  that  cj 
itself  moves  in  the  plane  of  ej  and  cg  from  the  initial  to  the 
terminal  point  of  cg- 

Evidently  e^ei,  interpreted  in  the  same  way,  gives  a  genera- 
tion of  the  same  parallelogram  in  the  reverse  sense,  and  should 
therefore  be  the  negative  of  ejCg. 

Similarly,  if  pi  and  pa  a^re  two  unit  points,  jhP-z  is  that  which 
is  generated  by  pi  in  moving  from  its  position  to  that  of  p2  in 
a  right  line ;  thus  piP2  is  a  line  of  definite  magnitude,  direc- 
tion, and  position ;  i.e.  a  point-vector,  according  to  Art.  1. 
Evidently  PiPi  =  —PiP^- 

Such  products  as  the  above  are  called  combinatory,  because 
the  factors  combine  to  form  a  new  geometric  quantity  different 
from  either  of  the  component  factors. 

15.  The  term  posited  will  be  applied  to  such  geometric 
quantities  as  have  definite  positions ;  as,  for  instance,  a  right 
line  or  plane  passing  through  a  definite  point  not  at  qc. 


24  DIRECTIONAL  CALCULUS.  [Art.  16. 

Any  two  geometric  quantities  which  differ  only  in  magnitude 
are  said  to  be  congruent. 

16.  Another  conception  of  combinatory  products  of  posited 
quantities  will  now  be  given,  which  will  be  found  to  be  con- 
sistent with  that  given  in  Art.  14. 

(a)  The  product  of  two  posited  quantities  which  have  no  com- 
mon figure  is  some  multiple  of  the  connecting  Jigure. 

(6)  TJie  product  of  two  posited  quantities  tchich  must  have  a 
common  Jigure  in  the  space  under  consideration,  is  the  common 
Jigure,  multiplied  by  a  scalar  quantity. 

Examples.  —  Under  (a),  the  product  of  two  points  is  the 
connecting  straight  line  as  in  Art.  14. 

Under  (b),  the  product  of  two  point-plane-vectors,  which 
must  necessarily  have  a  common  line,  is  that  common  line 
(point-vector)  multiplied  by  a  scalar  to  be  determined  here- 
after. 

(c)  A  continued  product  of  several  posited  quantities  is  to  be 
interpreted  by  taking  together  first  the  two  factors  on  the  right, 
then  the  result  of  this  midtiplication  with  the  next  factor  towards 
the  left,  then  this  result  ivith  the  next,  etc. ;  but  if  at  any  stage  of 
the  process  the  product  of  the  factors  treated  up  to  this  point 
becom£s  a  scalar  quantity,  then  this  scalar  will  not  form  a  com- 
binatory prodicct  with  the  next  factor  to  the  left,  but  is  to  be 
treated  like  any  other  merely  numerical  factor,  obeying  as  a 
WHOLE  the  laws  of  ordinary  algebraic  midtiplication. 

This  statement  cannot  well  be  illustrated  here,  but  its  mean- 
ing will  appear  in  the  sequel. 

Grassmann  calls  a  product  of  the  kind  (a)  progressive,  be- 
cause it  is  of  a  geometric  order  higher  than  that  of  either 
factor ;  while  one  of  the  kind  (6)  is  regressive,  because  it  is 
of  lower  order  than  either  factor.  If  in  a  continued  product 
of  factors,  as  in  (c) ,  some  of  the  successive  products  are  pro- 
gressive and  some  regressive,  then  the  product  as  a  whole  is 
said  to  be  mixed. 


Chap.  II.]  MTTLTIPLICATTON.  25 

17.  Lmcs  of  combinatory  multiplication  of  any  number  oj 
points  or  vectors  not  exceeding  the  number  of  ixdepexdent 
points  or  vectors  possible  in  the  space  under  consideration. 

These  are : 

The  associative  law ; 

The  distributive  law ; 

PliP2+P3)=lhP-2+PlPs (34) 

The  alternative  law ; 

PlP2=—P2Pl (35) 

If  2)2=  Pi  ill  (35),  we  have  2hPi  =  ^,  which  agrees  with  the 
meaning  of  the  product  of  two  points  as  given  in  Art.  14. 

Also  P1P2P1  =  Pi  •  P2P1  =  -Pi-PiP2  =  -PiPi-P2  =  0, 

by  (33)  and  (35)  ;  hence,  if  there  occur  two  identical  factors  in 
a  product  of  the  kind  stated  at  the  head  of  this  article,  the  product 
is  zero. 

The  meaning  of  certain  products  will  be  different  according 
as  they  are  interpreted  in  two-  or  three-dimensional  space ; 
hence  we  shall  apply  the  term  planimetric  to  such  a  product 
when  it  is  to  be  taken  in  two-dimensional  space,  and  the  term 
stereometric  when  it  is  to  be  taken  in  three-dimensional  space. 

"We  proceed  now  to  a  detailed  discussion  of  all  the  geometric 
products  possible  in  two-  and  three-dimensional  space. 

18.  Product  of  two  points.  Let  pi  and  pg  ^  two  unit  points. 
Their  product,  P1P2,  has  already,  in  Art.  14, 

been  stated  to  be  the  portion  of  the  right  line  p, ^ p* 

fixed  by  pi  and  pa  extending  front  2^1  to  p^  and 

it  will  be  called  a  point-vector,  or  simply  a  line  for  brevity. 

Let  €=P2—Pi'i  then 

2hiP2-lh)=Pl^  =  PlP2-PlPl=PlP2,       .       •       (36) 

by  the  last  article ;  so  that  the  product  of  two  points  is  equivar 
lent  to  that  of  the  first  point  into  the  vector  from  the  first  to 


26  DIRECTIONAL  CALCULtTS.  [Art.  19. 

the  second.  Hence  multiplying  a  vector  by  a  point  changes 
it  into  a  point- vector ;  i.e.  fixes  its  position.  Let  c'  be  any 
vector  whatever;  then,  by  eq.  (2),  pi  +  X€'  may  be  any  point 
in  space  by  suitably  choosing  x  and  c' :  multiply  into  e ; 
therefore 

(j?i4-xe')£=i?i£-f-rc£'£ (.37) 

Hence  the  point-vector  obtained  by  multiplying  the  vector  € 
by  the  point  2h  +  ^«'  is  not  in  general  equal  to  j)i£,  but  differs 
from  it  by  the  quantity  xe't.     If,  however,  we  have  £*  =  e,  then 

(Pi+X€)e=pi€  +  xe€=pi€', (38) 

so  that,  when  £  is  multiplied  by  any  point  on  the  line  through 
Pi  and  P2,  the  resulting  point-vector  is  equal  to  ^9|£. 

Hence,  in  order  that  two  point-vectors  may  be  equal,  they 
must  have  the  same  length,  the  same  direction,  and  must  be 
situated  upon  the  same  straight  line,  while  their  position  on 
this  line  is  indifferent. 

19.  Product  of  two  vectors.  Let  £1  and  £2  be  any  two  vec- 
tors :  their  product  ejEj  has  already,  in  Art.  14,  been  stated  to 
be  a  plane  area  ||  to  £1  and  £2  and  equal  to  the  area  of  the  par- 
allelogram whose  sides  are  parallel  and  equal  in  length  to  £1 
and  cg.     Write 

£  =  ccj£j  +  a;2€2  and   £'  =  r/i£i  +  y^2  5 

then  £  and  £'  may  be  any  two  vectors  ||  to  the  plane-vector  £i£2, 
by  giving  suitable  values  to  Xi,  x^,  yi,  y^.     Multiplying,  we  have 


«'  =  {^if-i+x^^i)  iVi^i+y^o)  =  (a-i?/2-a^i)£i£2  = 


X2  2/2 


£,£2.     (39) 


Hence  the  product  of  any  two  vectors  ||  to  £i€2  only  differs 
from  £i€2  by  a  scalar  factor.  If  ajjyj— .a^22/i  =  1?  then  ££'  =  £i£2. 
Thus  in  order  that  two  plane-vectors  should  be  equal,  it  is 
only  necessary  that  their  plane  directions  and  areas  should  be 
the  same,  without  regard  to  the  directions  of  the  component 
vectors.     Of  course  direction  as  applied  to  a  plane-vector  must 


Chap.  IT.]  jnJLTIPLICATION.  27 

include  the  sense  in  which  the  generation  takes  place.     See 
Art.  14. 

Regarding  cj  and  cg  ^s  points  at  oc,  "\ve  see  that,  just  as  a 
zero  point  at  x  is  a  vector,  so  the  product  of  two  such,  that  is 
a  point-vector  at  oo,  is  a  plane-vector ;  or,  in  other  words,  just 
as  a  point  at  oo  gives  a  line  direction,  so  a  line  at  oo  gives  a 
plane  direction. 

20.  Product  of  three  points.  Let  p„  p^,  p^  be  three  unit 
points.  By  Art.  16,  (a)  and  (c),  the  product  should  be  a 
multiple  of  the  connecting  tri- 
angle whose  vertices  are  ji\}  Ps,  Ps- 
By  Art.  17  the  product  obeys  the 
associative  law,  so  that 


n. 


PlP2P3=PlP2-P3-  P^' 

Hence,  by  Art.  14,  the  product  is  what  is  generated  bj'  the 
point- vector  p^p^  in  moving,  ||  to  itself,  in  the  plane  of  the 
three  points,  from  its  original  position,  till  it  passes  through 
p.^;  that  is,  a.  j^arallelogram  whose  area  is  ticice  that  of  the  con- 
necting triangle.     Let  p^  —jh  =  c,  and  P3—Pi  =  ^'',  then 

PlP2P3=PlP<'=Pl^^' (39) 

The  product  is  thus  a  posited  and  directed  plane  area  of  given 
magnitude;  that  is,  a  point-plane-vector,  or  simply  a  plane  for 
brevity,  especially  when  the  magnitude  is  a  matter  of  indiffer- 
ence. Eq.  (39)  shows  that  multiplying  a  plane-vector  by  a 
point  fixes  its  position  by  making  it  pass  through  the  point, 
since  by  Art.  14,  cc'  is  a  plane-vector.  Let  e"  be  a  vector  in 
any  direction  ;  then,  as  in  Art.  18,  p,  -}-  are"  may  be  any  point 
in  space.     Xow 

(p,  +  a;c")££'=p,€£'+X£"cc'; (40) 

so  that  the  point-plane-vector  obtained  by  multiplying  ££'  by 
the  point  pi  +  xe"  is,  in  general,  different  from  Pi£c'.  If,  how- 
ever, we  have  e"  =  ye-\-  ze',  so  that  t"  is  ||  to  ££',  and  pi  -f-  xt"  is 
a  point  of  the  plane  p\P-2ps,  then 

{Pi  +  »e")a'  =  (pi  4-  x{yi  -\-  zd)  )ec'  =i>iee' .     (41) 


28 


DIRECTIONAL   CALCULUS. 


[Art.  2L 


Hence  two  point-plane-vectors  are  equal  when  they  have  the 
same  area  and  direction,  or  sense,  and  lie  in  the  same  plane, 
without  regard  to  position  in  that  plane. 

21.  Product  of  three  vectors.  Let  cj,  cg,  cg  be  any  three 
vectors  not  ||  to  the  same  plane ;  then  €i€.2e3  =  ci£2  •  cg,  and,  by 
Art.  14,  the  product  is  the  parallelo-  p 
piped  generated  by  the  plane-vector 
ejCg  ^  it  moves,  ||  to  itself,  from  the 
initial  to   the    terminal   point   of  cg. 

Let       c  =  Xi€i  -j-  cCafg  +  ^3«3  =  2ia:e, 

then  £,  c',  c"  may  be  any  three  vectors  whatever  with  suitable 
values  of  the  scalar  coefficients,  and 


ce'c"  =  SxcS^/cSzc  = 


Xi 

a;2 

a^al 

Vi  2/2  ysl 

Zl 

22 

%l 

«lC2€3- 


(42) 


Hence  tioo  triple  products  of  vectors  can  only  differ  in  magni- 
tude and  sign,  and  two  such  products  ivill  be  equal  tchen  their 
magnitudes  and  order  of  generation,  or  sense,  are  the  same. 

It  follows  therefore  that  the  combinatory  product  of  three 
vectors  is  always  a  scalar  quantity,  by  the  definition  given  in 
Art.  1.  If  the  three  vectors  are  parallel  to  one  plane,  the 
volume  of  the  parallelopiped  becomes  zero,  and  the  product 
therefore  vanishes.  Hence  the  planimetric  comhinatoi'y  product 
of  three  vectors  is  always  zero.  If  we  regard  tj,  cj,  eg  as  points 
at  00,  we  see  that  a  point-plane-vector  at  co  is  equivalent  to  a 
solid,  which  carries  out  the  analogy  mentioned  at  the  end  of 
Art.  19. 


*  For  the  benefit  of  any  reader  who  may  not  be  familiar  with  deter- 


minants, it  may  be  stated  that  the  coefficient  of 


in  (42)  is  an  ab- 


breviated way  of  writing 

'1(3/2^3  -  ^3^2)  +  ^2(i'3«l  -  3/1^3)  +  ^3(^1-2  -  V'^l)^ 

which  expression  will  be  obtained  by  multiplying  out  the  values  of 
and  remembering  that  terms  containing  repeated  factors  vanish. 


Chap.  IT.]  MULTIPLICATION.  29 

22.  Product  of  four  j)oints.  Let  pi,  p2,  Ps,  p^  be  four  unit 
points ;  then,  by  Arts.  17  and  14,  we  have 

and  the  product  is  the  volume  generated  by  the  point-plane- 
vector  2hPiPi  when  it  is  moved  |(  to  itself  from  its  initial  posi- 
tion till  it  passes  through  p^ ;  that  is,  b,  paraUelopiped,  of  which 
P1P2,  IhPsi  ^nd  pip^  are  three  conterminous  edges.  This  volume 
is  six  times  the  connecting  tetrahedron  of  the  four  points, 
which  accords  with  Art.  16.     (See   figure  of  last  Art.)     If 

P2-P1  =  c,  Ps  -Pi=  «',  and  p^-p^  =  c", 
then  PiP-2PsPi=PilhPs^"  =lhPi^'^"  =Pi^^'^"-  •     •     (43) 

The  point  pi  -f-  a^  -f-  ye'  +  zc"  may  be  any  point  whatever, 
with  suitable  values  of  x,  y,  z,  and  we  have 

(p,-fa^  +  ye'  +  2£")«'£"=i9,«V',      ...     (44) 

so  that  any  point  whatever  may  be  substituted  for  pi  in  (43) 
without  changing  the  value.  By  the  above,  and  by  the  pre- 
ceding article,  it  appears  that  a  product  of  four  points  is  equal 
to  the  product  of  any  other  four  points  having  the  same  mag- 
nitude and  intrinsic  sign,  or  order  of  generation :  thus  such  a 
product  is  a  scalar;  and,  in  fact,  differs  in  no  manner  what- 
ever from  a  product  of  three  vectors  which  has  the  same  mag- 
nitude and  sign.     "We  may  therefore  write 

p,ce'c"  =  «V' (45) 

If  the  four  points  are  in  one  plane,  the  product  is  zero, 
because  the  volume  of  the  connecting  tetrahedron  is  zero. 
Thus  the  planimetric  progressive  product  of  four  points  is 
always  zero. 

23.  The  stereometric,  progressive  product  of  four  or  more 
vectors  is  always  zero.  But  we  may  have  cj  •  cgCac^,  meaning 
the  vector  e,  times  the  scalar  co^s^*?  etc. 

The  stereometric,  progressive  product  of  five  or  more  points  is 
always  zero.   But  we  may  have  such  products  as  pip^  'PzPiPiPo 


80  DIRECTIOXAL  CALCULUS.  [Art.  24. 

meaning  the  ordinary  algebraic  product  of   the  point-vector 
PiP2  into  the  scalar  2hPiP.'>P6- 

24.  From  the  preceding  articles  we  have  the  following 
conditions : 

lhlh  =  ^ (46) 

is  the  condition  that  two  points  shall  coincide. 

PiP2Ps  =  0 (47) 

is  the  condition  that  three  points  shall  be  in  one  right  line. 

■P^PiP^P^  =  ^ (48) 

is  the  condition  that  four  points  shall  be  in  one  plane. 

£,£.  =  0 (49) 

is  the  condition  that  two  vectors  shall  be  parallel. 

£l£2£3=0 (50) 

is  the  condition  that  three  vectors  shall  be  parallel  to  one 
plane. 

In  the  further  development  of  the  subject  it  will  be  con- 
venient to  treat  separately  two-  and  three-dimensional  space, 
considering  the  former  first. 

Plaximetric  Products. 

25.  In  two-dimensional,  or  plane,  space  two  plane- vectors, 
or  two  point-plane-vectors,  cannot  differ  from  each  other  except 
in  magnitude  and  sign,  since  both  are  restricted  to  one  plane. 

Hence  they  become  scalar  quantities.  Furthermore,  by 
Arts.  19  and  20,  the  product  of  two  vectors  is  now  identical 
in  meaning  with  the  product  of  three  points.  Thus,  if  P2—Pi=(^ 
and  Pz—P\  =  c'j  we  have,  in  plane  space, 

P,££'  =  ££' (51) 

Plane  space  is  the  locus  of  all  points  dependent  on  three 
fixed  reference  points.  We  shall  call  these  e^,  e,,  e^,  and  shall 
always  take  the  area  e^^e^  as  the  unit  of  measure  of  area,  when 


Chap.  H.] 


MULTIPLICATION. 


31 


dealing  with  a  point  system.    That  is,  we  write  always  in  plane 
space, 

eoe^e,^! (52) 

This  is  a  great  practical  convenience,  and  in  no  way  affects  the 
generality  of  results. 

The  sides  of  the  reference  triangle  taken  around  in  order  are 
called  the  complements  of  the  opposite  vertices;  thus 

gje,  =  (complement  of  ^o)  =  l^o 

62^0  =  (complement  of  ei)  =  |ei  V,  .     .     .     .     (53) 

e^ei  =  (complement  of  e,)  =  'e.2 

the  vertical  line  before  the  point  being  called  the  sign  of  the 
complement. 

In  dealing  with  a  vector  system  we  shall  usually  refer  to  a 
unit  normal  system  of  vectors  ij,  12,  as  stated  in  Art.  12.  We 
shall  then  have  in  jilane  sjjoce 

'1^2=1 (54) 

26.   Let 

2h  =  k^o  +  ^1^1  +  i-A2  =  ^e,  p.,  =  Xme,  p.  =  ^ne ; 

then        ih  Pi  =  .1"  .1'  eoe,  +  l'  2  e^e^  +  J  J  e^o 


Wo  7711 


__      to      {j       12 

~"|!mo  mi  mo 


6061  + 


I,       h 

nio  wio 

Co   |Cl    1^2 
to      tl      '2 

w-o  mi  ???2 


(55) 


The  third  and  fourth  members  of  (55)  are  simply  different 
ways  of  expressing  the  second.  The  fourth  member  is  espe- 
cially noticeable  for  its  symmetry.  Eq.  (55)  shows  that  any 
point-vector  is  expressible  in  terms  of  the  sides  of  the  reference 
triangle. 

tfl     'i     ^2 

Again,    P\Pilh  =  2/e  •  2me  •  2ne  =  mo  m^  mj  ;  .     .     .     (56) 

Wo      «!     Wo 

as  will  be  found  on  multiplying  out  and  putting  eyfiie.^  =  1,  by 
(52). 


32  DIKECTIONAL  CALCULUS.  [Art.  27. 

27.  Exercises.  —  1.  Find  by  eq.  (47)  the  condition  that 
the  three  points  p  +  cj,  p  +  e.,,  p  +  «:;?  shall  be  collinear,  and 
illustrate  geometrically.  Aiis.  cjCo  +  £2C3  +  csEi  =  0. 

2.  Show  that  if  a  line  be  expressed  in  terms  of  the  sides  of 
the  reference  triangle,  and  the  sum  of  the  coefficients  be  zero, 
the  line  passes  through  the  mean  point  of  the  reference  points. 

28.  Since  the  product  of  three  points  obeys  the  associative 
law,  it  can  be  regarded  as  the  product  of  a  point  into  a  point- 
vector,  or  of  a  point-vector  into  a  point.     Thus,  if  L  —ihlh^ 

PlP2P3=Pl-P2Ps=PlL=lhP3Pl  =  L2h,      '       .      .        (57) 

so  that  this  product  is  commutative. 

29.  Product  of  two  lines,  or  point-vectors.  The  products 
hitherto  considered  have  all  been  progressive  ;  we  now  come  to 
one  which  is  regressive.  Since  two  lines  in  plane  space  must 
intersect,  they  come  under  Art.  16,  (6).   Let  

the  lines  be  L^  and  Lo,  let  Pq  be  their  com-  f^i  / 

mon  point,  and  let  pi  and  p.2  be  so  taken  that  /  / 

Li=PoPi,  and  L.^PoPi.  / 

We  may  also  write,  /_ / 

Li=Po{Pi-Po),  L2  =  Po{p2-Po);       '       ^' 

now  the  product  of  the  vectors  (Pi—Po)  (po  —Po)  is  the  area  of 
the  parallelogram  on  these  vectors,  and  is  scalar ;  the  product 
of  the  point-vectors  should  certainly  give  this  result,  and,  in 
addition,  the  point  fixed  by  them,  viz.  their  intersection.  This 
is  in  accordance  with  Art.  16,  (&).     The  product 

(Pl-Po)(P2-Po) 

is  equivalent  to  P0P1P2 ',  hence  we  may  write 

LiL2=PoPi-PoP2=2^oPiP2-Po (»8) 

It  is  to  be  carefully  noted  that  the  third  member  of  eq.  (58)  is 
not  derived  from  the  second  by  interchanges,  according  to  the  asso- 
ciative and  alternative  laws,  but  is  an   independent  expression. 


Chap.  II.]  MULTIPLICATION.  33 

which  gives  the  meaning  of  the  product  of  two  lines.  It  may  be 
regarded  as  a  model  form  for  the  treatment  of  regressive  prod- 
ucts. Thus,  if  AB  and  AC  are  any  two  quantities  whose 
product  is  regressive,  and  if  A  is  their  common  figure,  we 
shall  always  have 

AB-AC=ABCA (59) 

We  have  accordingly 

LoLi  =  P0P2  •  PnPi  =  IMhlh  •  Po  =  -  IhPilh  'Po  =  -  L1L2,  (60) 
or  the  product  of  two  lines  is  non-commutative. 

30.  Product  of  a  point  and  two  lines.  Let  Xj  and  L2  be  as 
in  the  last  article,  and  p  be  some  point ;  then 

pLiLo  =  p  ■  PoPi  •  2hP'2  =  P  •  l^olhlh  •  Po  =  l^oPilh '  PPa-      ■     (61) 

P0P1P2  can  be  placed  first  because  it  is  a  scalar.  This  is  a 
mixed  product,  that  of  the  two  lines  being  regressive,  and  that 
of  p  into  their  common  point,  py,  being  progressive.     Also, 

pLiL.,  =  -2iL>Li  =  L.,Li-p (62) 

The  period  is  necessary  in  the  last  member  of  this  equation 
to  preserve  the  meaning ;  that  is,  the  product  of  the  points 
L1L2  and  p.  Without  the  period  the  expression  would  mean 
the  line  L2  multiplied  into  the  scalar  Xjp. 

31.  Product  of  three  lines.  Let  Li,  L2,  L^he  three  lines,  and 
P\j  P21  Ps  their  common  points,  and  take  scalar  factors  rij,  w,,  n^ 
so  that  Li  =  7ii2)2P3i  L2  =  n22hP\i  ^-^s  =  '>h2hP-2]  then 

LiLzLs  =  niWorigi^oPa  •  PsPi  -PiP'i  =  -  nii^2ihPip3  •  P\P^  •  P1P2 

=  -nin2nsP2P3-PiP3P2-Pi  =  nin2ns{PiP2P3y.     .     (63) 

It  thus  appears  that  in  plane  space  lines  obey  the  same  laws 
of  multiplication  as  points. 

32.  Let  two  points  be  given  each  as  the  common  point  of 
two  lines,  viz.  pi  =  LgLi  and  P2  =  L0L2,  then 

P1P2  =  LoLi '  L^Li  =  L0L1L2 '  Lo, (64) 


34  DIKECTIONAL  CALCULUS.  [Art.  33. 

which  is  a  reciprocal  equation  to  (58).     Similarly,  if 

2)i  =  wiiLoLg,  po  =  vioL^Li,  2h  =  w^s^iA, 
we  have 

P1P2P3  =  WimamgLoiyg  •  L^Li  -  LyL.,  =  mimim^^LiL-iL^y.      (65) 

33.  The  condition 

A-Z'2  =  0 (66) 

requires  that  Xj  and  L.2  shall  be  congruent,  that  is,  be  situated 
on  the  same  right  line;  for,  by  (58),  this  gives  PoP\P2  =  ^- 
The  condition 

A  A'^3  =  0     .     .  • (67) 

requires  that  the  three  lines  shall  pass  through  a  common 
point;  for  LiL^  is  some  point,  say  p,  and  Lip  =  0  makes  L^ 
pass  through  p,  by  eq.  (47). 

34.  Product  of  parallel  lines.     Let  the  lines  be  Li  =  pit  and 
L2=-np^\  then 

L1L2  =  npif.  'p.^  =  ntp^  •  e/?2  =  nepipo  •  c  =  np^j^of^  •  e.  .     .     (68) 

Thus  the  product  is  the  common  vector,  or  point  at  00,  mul- 
tiplied by  the  scalar  npip^c. 

35.  Product  of  two  vectors.     Let 

£j  =  wijij  -f-  m.j.-,,  £9  =  '^I'l  +  ^2*? ; 
then,  since  iii,  =  1, 

_|mi  ^Ual  ^gQ 

But,  since  eiCj  is  a  parallelogram  whose  sides  are  ||  to  cj  and 
Co,  we  have  m  ti     •     ^^1 


Also,    mj  =  Ttx  cos  <  ^'  =  Tti  cos  a^,  say ;    m<,  —  Tcj  sin  oi  j 
rii  =  Tu  cos  02 ;    »2  =  ^^2  sin  U2. 


Chap.  II.]  MULTIPLICATION.  35 

Therefore 

which  affords  a  proof  of  the  trigonometrical  formula  for  the 
sine  of  the  difference  of  two  angles. 

36.  Sum  of  point-vectors.  Using  Li,  Lj,  po,  pi,  jh  a^s  in  Art. 
29,  we  have 

Li  +  Lo=^2Mh  +P0P2  =l>o(Pi  +i>2)  =  ^PoP,  .     .     ('1) 

if  p  is  the  mean  point  of  2h  and  />,•  Thus,  Zj  -|-  Lo  passes 
through  the  common  point  of  the  two  lines  and  is  equal  in 
length  to  that  diagonal  of  the  parallelogram  on  Li  and  L., 
which  passes  through  j:),,.     Similarly, 

Li-L.,=p^{Pi-2)2); (72) 

so  that  the  difference  of  Li  and  L.,  passes  through  po,  and  is 
equal  in  length  to  the  other  diagonal  of  the  parallelogram. 

Suppose  Li  and  Lo  to  be  jjarallel,  and  equal  respectively  to 
npi€  and  2h^ ',  then 

Xi  +  A>  =  («i^i+P2)c  =  ('t  +  l)i>c;       ....     (73) 

that  is,  the  sum  is  a  ||  point- vector  passing  through  the  mean 
point  of  npi  and  p,.     Finally,  let  n  =  —  1,  so  that 

L,+L.,  =  {2h-2h)^, (74) 

and  the  sum  is,  in  this  case,  the  product  of  two  vectors ;  that 
is,  in  plane  space,  a  scalar. 

The  reader  will  notice  the  exact  correspondence  between 
the  results  of  this  article  and  the  resultant  of  forces  in  plane 
space. 

37.  Sum  of  sides  of  a  polygon.  Let  1,  2,  3  be  any  three 
points;  then 

12  +  23  +  31  =  23  -  21  +  11  -  13  =  2(3  -  1)  -  1(3-1) 

=  (2-l)(3-l). 


36  DIRECTIONAL  CALCULUS.  [Art.  38. 

Hence  the  sum  of  the  three  point-vector  sides  of  a  triangle 
taken  around  in  order  is  equal  to  twice  the  area  of  the  triangle. 
Similarly,  let  1,  2,  3,  4,  •  •  •  n,  be  n  points,  the  vertices  of  a  poly- 
gon of  n  sides ;  then 


12  +  23  4-  31  =  (2  -  1)  (3  -  1) 
13 +  34 +  41  =  (3-1)  (4-1) 


l(ji  -  1)  +  (n  -  l)n  +  nl  =  [(h  -  1)-  1]  (n  -  1) 
and,  adding, 

12  +  23  -I-  34  +  •••  +  nl  =  twice  the  area  of  the  polygon. 

38.  Complement.  —  (a)  The  complement  of  a  reference  unit 
is  the  product  of  the  other  reference  units,  so  taken  that  theprodvx^t 
of  the  unit  into  its  complement  shall  be  positive  unity. 

This  definition  is  perfectly  general,  and  applies  to  either  a 
point  or  vector  system  in  space  of  any  number  of  dimensions. 
We  have  already  had  examples  in  eq.  (53,)  as  |eo  =  eiC^  whence 

(b)  The  complement  of  a  scalar  quantity  is  the  quantity  itself. 
Thus,  \n  =  n;    lijij  =  11I2  =  1. 

(c)  The  complement  of  the  product  of  several  factors  is  equal 
to  the  product  of  the  complements  of  the  factors. 

Thus,  \{np)  =  \n\p==n\p;    Iiii2  =  ki|t2. 

(d)  The  complement  of  the  sum  of  several  quantities  is  the  sum 
of  the  complements  of  the  quantities. 

39.  Complement  in  a  plane  vector  system.  Taking  a  unit 
normal  system  ij,  i^  we  have,  according  to  the  previous  article, 

Complement  of  i^,  written  ]ii  =  12,        for  ij[ii  =  ijig  =  1 ; 
Complement  of  t^  written  [ij  =  —  ii,  for  ijltj  =  — ijii=titj=l  j 

|]ii  =  I12  =  — .ii ;  l!t3  =  —  !'i  =  —h' 


Chap.  II.] 


MULTIPLICATION. 


37 


Let  cj  =  iriiii  +  ?)i2i2 

and  €2=  »i  ti  +  ^212 ;  then 

|ci  =  rn-ilii  +  mi\i2  =  mi^  —  wigti.     (75) 

By  the  figure  it  is  evident  that  jci  is  S  | 
a  vector  of  the  same  length  as  c^,  and 
perpendicular  to  it,  or,  in  other  words, 
taking  the  complement  of  a  vector  in  plane  space  rotates  it  jwsi- 
tively  through  90°. 

The  product  e^ko  is  the  parallelogram  whose  sides  are  e^  and 
[cj;  if  ci  is  parallel  to  \£^  the  area  of  the  parallelogram  van- 
ishes, or  Ci[c2  =  0 ;  but,  since  [cj  is  X  to  cj,  tj  must,  in  this  case, 
be  X  to  c, ;  hence  the  equation 

Cl!£2  =  0 (76) 

is  tlie  condition  thai  the  two  vectors  ci  and  t^  shall  be  jjerpendic- 
ular. 

The  product  cjlci  is  the  area  of  a  square  each  side  of  which 
is  of  the  length  Tci ;  hence 

ei|ei=r-£i==£i?,  say (77) 

The  form  tp  is  merely  another  way  of  writing  ei\t^  which  is 
often  convenient.     If  n  is  a  scalar,  we  have 

n\n  =  nn  =  w-, 

whence  the  analogy  is  apparent. 

Grassmann  calls  Ci|e2  the  "  inner  product "  of  cj  and  eg?  regard- 
ing the  complement  sign  as  a  species  of  multiplication  sign,  and 
accordingly  calls  ^  the  "  inner  square  "  of  e.  It  seems  prefer- 
able to  the  author  not  to  introduce  a  new  species  of  multipli- 
cation, but  to  regard  ei|C2  as  simply  the  combinatory  product 
of  ci  into  I  £2,  a  way  of  looking  at  it  which  is  practically  far 
more  simple,  and  renders  interpretation  easier.  Somewhat 
after  the  analogy  of  the  word  cosine  for  sine  of  the  complement, 
we  may  call  ^  the  co-square  of  c,  and,  just  as  we  read  a*,  a 
square,  we  may  read  ^,  c  co-square.  Similarly,  fjU,  may  be 
called  the  co-product  of  cj  and  cg- 

48G07 


38  DIKECTIONAL   CALCULUS.  [Art.  39. 

We  have  also, 

We  obtain  the  third  member  because 

and  i2|t2  =  ~  hh  —  hh  =  1  > 

the  fourth  member  is  apparent  from  the  symmetry  of  the  third. 
It  will  be  found  that  Ave  have  always,  when  A  and  11  are  quan- 
tities of  the  same  order  in  the  reference  units, 

A\B  =  B\A (79) 

Since  Tie  =  Tje,  as  shown  above,  we  have,  as  in  Art.  35, 
ci!c2  =  Tti  Te.,  X  sin  (ang.  bet.  cj  and  Ic)  =  Tt^  Tc,  cos  < ^\     (80) 
Also,  taking  the  values  of  mj,  m^,  th,  n^  as  in  Art.  35,  we  have 
ci]c2  =  m,Wi  +  moTt-i  =  Tti  7^2 (cos  aj  cos  a.,  +  sin  ax  sin  a.^ 

=  r€ire2Cos<^ (81) 

which  affords  a  proof  of  the  trigonometrical  formula  for  the 
cosine  of  the  difference  of  two  angles. 
If  (.2  =  ^15  we  have 

.-.   Tt,=  +^m^^  +  mi   I ^  "^ 

Square  and  add  eqs.  (70)  and  (80)  ;  therefore 

r^r%  =  exV  =  (eic.,)^+(cilc2)i  .    .    .    (83) 

If  t  be  any  unit  vector,  and  p  any  other  vector,  we  have 
t .  p|i  =  I .  Tp  •  cos<''  =  projection  of  p  on  direction  of  i, 

and  t;o  •  Ji  =|i  •  Tp-  sin<^  =  projection  of  p  on  direction  _L  to  i; 

hence  we  may  write 

p  =  i-p\i  +  ip'\L (84) 

This  equation  may  be  verified  by  multiplying  successively 
by  I  and  |i. 


Chap.  IL]  MULTIPLICATION.  39 

40.  Complement  in  a  plane  jioint  system.  Taking  as  reference 
points  Co,  Ci,  62,  with  the  condition  Cyeigg  =  1,  we  have,  in  accord- 
ance with  Art.  38, 


(85) 


\ei  =  e._A),         16260  =  1161  =  61 

|62  =  6o6„  1^061=1162  =  62 

1(606162)  =leo  •  lei .  je.,   by  Art.  38,  (c), 

=  616,  •  6,60  •  6061  =  (606169)^  =  1  =  606162  - 

Note  that  the  complement  of  the  complement  of  a  reference 
point  is  here  the  point  itself,  while  in  the  vector  system  the 
complement  of  the  complement  was  negative.  The  general  law 
is  that  when  the  number  of  reference  units  is  even,  the  com- 
plement of  the  complement  of  one  of  them  is  negative,  and 
when  the  number  is  odd,  it  is  positive. 

Let  pi  =  IqCq  -j-  ^161  +  ^2623  and  p.,  =  ?noeo  f  miCi  +  wi'262  be  any 
two  points ;  then 

|Pl  =  Itih  -f  /l'6i  +  Z2162  =  ^06162  +  ^16260  +  ^26061 

=  Y  (^061  -  ^160)  (^062  -  ^260)  ;      .     (86) 

so  that  the  complement  of  any  point  is  a  point-vector.  The 
fourth  member  of  (86)  expresses  this  point-vector  as  the  prod- 
uct of  the  points  in  which  it  cuts  two  of  the  sides  of  the  refer- 
ence triangle,  so  that  it  may  be  easily  constructed.  We  have 
also 

PilPi  =  (^060  +  hei  +  ^262)  {moCie^  -f  ?«,,62''o  +  ^"26061) 

=  If/mo -\-limi-^l2m2=p2\Pi, (87) 

so  that  eq.  (79)  is  here  verified.  Note  that  the  product  is 
scalar.     We  have,  however, 

^O^lj^i  =  6061  •  6260  =  —  696162  •  60  =  —  60  ">  /OQ\ 

A  I  I  ■(■}'-'         (00) 

and  6i\e(fii  =  6162  =  |eo  ) 

so  that,  when  the  quantities  on  each  side  of  the  complement 
sign  are  of  different  order,  the  product  is  not  scalar,  nor  commu- 
tative about  the  sign. 


40 


DIRECTIONAL   CALCULUS. 


[Akt.  4L 


If  Li  =  \pi  and  L2  =  |i\>,  we  have  jh  =  1 A  ^^^^  Ih  =  i  A>  ^^^ 
Li\L._=[jh  '  \\lh=^\pi  ■Ih=pM'*i  =  lh\lh  =  HL,,     (89) 
which  also  agrees  with  eq.  (79).     Again, 

LxL'i=\lh-\p-i  =  \PiP-z\ (90) 

so  that  the  common  point  of  the  lines  L^  and  L.,  is  the  comple- 
ment of  the  line  p^p-,.     Also, 

A|i>.  =  i!(i,!P2)  =  |(|A-i>.)=l(-i^.|A)  =  -|(i\>!i>,),  .    (91) 

of  which  eqs.  (88)  are  a  special  case. 

41.  Geometric  interpretation.  For  the  sake  of  simplicity, 
suppose  Cq)  6i>  ^2  to  be  the  cor-  ^ 

ners  of  an  equilateral  triangle. 
With    e,    the    mean    point    of  ^< 
the  triangle,  as  a  center,  draw  a 

circle  of  radius  — -,  a  being  the 

Vo 

side  of  the  triangle;  then,  p 
being  any  point  whatever,  \p  is 
its  anti-polar  with  reference  to 
this  circle ;  that  is,  a  line  par- 
allel to  its  polar,  and  equidistant 
with  it  from  the  center  of  the 
circle,  but  on  the  opposite  side. 

It  is  evident  that  with  reference  to  the  above  circle  each 
vertex  is  the  anti-pole  of  the  opposite  side  of  the  reference 
triangle ;  for  the  respective  distances  of  a  vertex  and  its  oppo- 
site side  from  the  center  are 


aV3 


and  - 


tV3 


2         Va  3        2         2V3' 

and  the  radiiis  of  the  circle  is  a  mean  proportional  between 
these.  The  figure  shows  the  construction  for  the  radius  of  the 
circle,  and  also  for  the  anti-polar  of  any  point  p. 

If  the  reference  triangle  is  not  equilateral,  it  can  be  obtained 
by  projection  from  an  equilateral  one,  and  the  circle  corre- 


Chap.  II.]  MULTIPLICATION.  41 

spending  to  that  will  be  projected  into  an  ellipse  such  that, 
with  reference  to  it,  \p  is  always  the  anti-polar  of  p.  Hence  it 
will  only  be  necessary  to  prove  the  property  in  the  case  of  an 
equilateral  reference  triangle.  Before  proceeding  to  the  proof, 
however,  it  will  be  necessary  to  find  an  expression  for  the  dis- 
tance between  two  points  in  terms  of  their  coefficients. 

42.  Distance  between  two  points.    Let  the  points  be  pi  and 
po  as  in  Art.  40.     Since  they  are  unit  points,  we  have 

2i?  =  1  =  S?m ;    .-.  Zo  =  1  —  ^1  —  h,   mn  =  1  —  Wj  —  mg. 
Hence 

p)^  =  2?e  =  e,,  +  ki^x  —  e,,)  +  kie-i  —  eo)  =  e„  -f  ^i^i  +  W  say, 
and  2h  =  2me  =  ^o  +  ^^i (^i  —  e,,)  +  m^ie.,  —  Co)  =  f,,  +  WiCj  +  mse^. 

Thus,  2h  —Pi=  (*^i  —  ^i)  «i  +  (wi2  —  lo)  C2, 

and  the  required  distance 


=  T{2h-Pr)  =  ^\.('m,-h)€,  +  {m,-h)c,Y-,    by  eq.  (82), 

=  ^(m,-k)W+  {m2-l.^\?+2{m,-h)  (m,-k)€,\e,.      (92) 

In  the  case  of  an  equilateral  reference  triangle,  whose  side 
is  a,  we  have  c^-  =  cj-  =  a^,  and  ei|c2  =  a^cos  60°  =  ^a^ ;  so  that 
eq.  (92)  becomes 


T(Po-Pi)  =aV(mi-Zi)'+  (^n,-L)-+{m,-h) (m^-k).    (93) 

43.  Proof  of  the  anti-polar  property  of  the  complement  in  a 
point  system.  Referring  to  the  figure  of  Art.  41,  we  propose 
to  show  that 

T{p  -  e)  T(e  -p')  =-  =  1^, 
6 

which  will  establish  the  proposition.    We  have,  since  p'  is  the 
common  point  of  |p  and  pe,  p'  congruent  with 

pe  '\p  =  xp  +  ye. 


42  DIRECTIONAL  CALCULUS.  [Art.  43. 

Multiply  both  members  by  ppx ;  then, 

PP\  'PC.  •  \p=pPie'P\p  =  ypp\e ; 

whence  y  =P'' 

Multiply  the  same  equation  by  piS ;  therefore 

p^e  . pe  '  \p  =  epxp  'e\p  =  xp^ep  =  —  xep^p  ; 

whence  x  =  —  e\p. 

Substituting,  we  have 

,^p^.e-e\p.p ^94^ 

2Ji-—p\e 

Hence  we  have 

,      _      p^-e  —  elp-p      €\p-(p  —  e) 

e  —p'  =  e  — ^ — ^ — ^  =   '^  „  ^    _, ' 

1^  —  e\  p  pp  —  e\  p 

and  T{p-e)T{e-p')  =  '\P'Y'^P-'^.   .     •     (95) 

^  p^  —  e\p 

Now,  by  eq.  (93),  taking  l^,  k,  k  as  the  coefficients  for^),  and 
Wo  =  mi  =  wig  =  ^  as  those  for  e,  we  have 

=  [h' -hlr  +  hk- 1,-1-2  + U<^'- 
Also, 

p?  =  pIp  =  l,^  +  Z,2  +  Z/  =  (1  _  Z^  _  ?,)2  +  ?j2  _^  12  » 

=  2(?i2  +  //-/i-Z2  +  ZiZ,  +  i), 

and  ?lp  =  K^o  +  ?i  +  Q  =  i, 

so  that 

Hence  (95)  becomes 
T(p-e)  Tie  -p')  =  ^  ""'^^^  +  ^^  +  ^^^^ "  ^'  "  ^^  +  J)  =«l  q.e.d. 


Chap.  TI.]  MULTIPLICATION.  43 

44.  The  conception  of  the  complement  in  a  point  system, 
as  developed  Arts.  40-43,  is  not  found  in  Grassmann's  works. 
He  deals  exclusively  with  the  complement  in  a  unit  normal 
vector  system.  See  Die  Ausdehnungslehre,  1862,  Art.  330.  In 
a  remark  at  the  end  of  Art.  337  he  shows  how  the  idea  of  the 
complement  might  be  extended  to  a  point  system,  but  in  a 
way  entirely  different  from  mine,  and  one  which  he  himself 
evidently  considered  of  no  practical  value,  since  he  has  made 
no  application  of  it.  On  the  contrary,  the  method  above  de- 
veloped is  of  great  utility,  giving  at  once  reciprocal  properties, 
according  to  the  principle  of  duality. 

In  a  subsequent  chapter  the  anti-polar  property  of  the  com- 
plement will  be  established  in  a  different  manner,  directly,  for 
a  reference  triangle  of  any  shape. 

45.  A  multiplication  table  for  a  point  system  in  plane  space 
is  given  on  page  44.  The  product  of  any  quantity  at  the  left 
into  any  at  the  top  is  found  at  the  intersection  of  the  corre- 
sponding row  and  column.  Thus,  e^o  •  ^i^i  =  —  e»  62I60  =  0?  etc. 
Algebraically  considered  this  system  forms  a  seven-fold  alge- 
bra, seven  reference  quantities  being  required  to  express  all 
quantities  of  the  system,  including  scalars. 

46.  Projections.     If  we  write  the  equation 

p  =  Xici -f  a^aco, (96) 

«iCi  is  evidently  the  projection  of  p  on  cj  ||  to  c,,  and  x-^e.,  is  the 
projection  of  p  on  c,  |(  to  cj.     Multiply  / 

(96)  into  £2 ;   therefore  pta  =  ^fy^  since  / 

p€.i  /  ^-/^ 

cjjCo  =  0 ;  thus,  Xi  =  — -.     Similarly,  mul-  /^  P. 

£l£2  ^    -^ 

tiplying  into  £1,  we  have  JCj  =  — - ;  whence   ^_ 

£1  •  peo        €.)  •  p£i 

p  =  -UL^  +  .^JL\     .......     (97) 

Hence  the  projections  of  p  on  q  ||  to  €2  and  on  £2  ||  to  ci  are 


44 


DIRECTIONAL  CALCULUS.  [Art.  46. 


A    MULTIPLICATION    TABLE     FOR     A     POINT     SYSTEM    IN 
PLANE    SPACE. 


^ 

11^ 

o 

rt 

N 

« 

^ 

o 

o^ 

w 

<a 

W 

i» 

<a 

la 

'"' 

o'^ 

<»'= 

o 

^ 

V 

II 

^ 

o 

o 

<^ 

1 

O 

14* 

«" 

<»" 

o" 

o 

<» 

"o 

O 

T-i 

o 

1 

o 

ii 

<a 

Sa 

U 

» 

iji" 

ll„ 

O 

o 

T-4 

O 

o 

1 

IW 

«r 

' 

tt*^ 

ftT' 

o 

« 

1 

^ 

o 

O 

o 

■i) 

o 

M 

«l 

«i 

«) 

o 

O 

o 

"i^ 

o 

1 

sa^ 

O 

o 

rH 

«'' 

Chap.  II.]  MULTIPLICATION.  45 

respectively  '-  and .     These  are  particular  cases  of 

a  general  proposition  which  may  be  stated,  as  follows. 

Let  £  be  a  quantity  of  the  nth  order  in  the  reference  units, 
and  C  one  of  the  mih.  order,  and  Ifet  the  number  of  reference 
units  be  vi  +  n,  so  that  BC  is  scalar :  then  the  projection  on  B 
of  any  quantity  A,  directed  by  C,  is 

B-AC 
BC 

Similarly,  the  projection  of  A  on'C,  directed  by  B,  is 

C-AB 
CB 

We  shall  have  also 

j^^B.AC     C-AB .ggv 

BC  CB  ^     ^ 

Since  we  have  restricted  ourselves  to  space  of  two  and  three 
dimensions,  we  shall  not  give  a  general  proof,  but  shall  verify 
and  explain  the  proposition  in  such  cases  as  arise  under  this 
restriction.  See  Die  Ausdehnungslehre,  1844,  Chap.  5,  and  the 
same,  1862,  §§  127-129. 

If  in  (97)  cu  C2  are  a  unit  normal  system,  replace  them  by  t^ 
and  i2 ;  then,  since  iiij  =  1, 

p  =  ii .  pt2  — 12  •  ph  =  ti  •  p|ii  +  hp  '  \h,      ....     (99) 
which  agrees  with  eq.  (84). 

47.   Consider  next  the  point  equation 

P  =  x„po  +  Xi2h-^^oP2; (100) 

multiply  successively  by  pip.21  P2P0,  and  poPi,  and  we  find 

_PPlP2  _PP2P0  _PPoPi 


'ihPxPi        ^      P1P2P0  P2P0P1' 

whence   p  =- [Po-PlhP2+Pi-PP2Po-\-P2-PPoPi]-      (101) 

PoPlPi 


46  DIRECTIONAL   CALCULUS.  [Art.  47. 

Each  term  in  the  brackets  taken  with  the  outside  factor  is 
of  the  typical  form  (98),  and  is  the  projection  of  p  on  one  of 
the  points  on  which  it  depends.     Write  again, 

P  =  !^\PlP2  +  XilPilh  +  XilPoPi, 
and  multiply  into  \po,  |pi,  \p2  successively,  and  we  find 

P  = l\PiP2 'P\Po-h\P2Pi>-p\Pi+\Pi>Pi •!>[ A>]-  (102) 

P0P1P2 

We  might  also  have  obtained  (102)  from  (101)  by  putting 
\P1P2  for  po,  etc.  The  terms  of  the  right-hand  member  of  (102) 
are  again  of  the  typical  form,  and  are  the  projections  of  p  on 
the  anti-poles  of  jhP-2i  P-iPni  ^^^d  PoPi-     Note  that 

\P\P2  •  li'o  =  1  (i>ii>2Po)  =  P1P2P0  =  P0P1P2' 

If  in  (101)  we  take  the  last  two  points  together,  their  sum 
is  some  point  on  p^p^  and  also  on  p)}')^^  since  p  is  expressed  in 
terms  of  po  ^-'^d  this  point.  Hence  this  point  is  congruent  with 
PiPi  •  PPoy  ^^^  we  may  write 

P  =  x])f,  +  yihP>  •  l>Po- 
Multiply  into  2h2h}  ^^fl  we  have 

PP1P2  =  ^0PlP2- 

Multiply  into  poPi,  and  we  have 

ppoPi  =  ypiP2  -ppo-PoPi  =  yihP2i^i,  -ppoPi, 
1 

or  y  = 

P0P1P2 

Substituting  values  of  x  and  y,  we  obtain 

P«-mP,    P,P,-m 

^         P0P1P2  IhPePo  '  ^        \ 

an  equation  of  the  same  form  as  (98).  The  second  term  of 
the  right-hand  member  of  (103)  is  the  projection  of  p  on  pip.^ 
directed  by  2>0' 


'""H.     .     .     (104) 


Chap.  II.]  MULTIPLICATION.  47 

The  second  members  of  (101)  and  (103)  must  be  identically 
equal ;  hence  we  have 

PaP2  -PP^  =Pi  -PPiPo  +2h  'PPoPu 
or,  writing  p^  instead  of  p,  and  j)i  instead  of  p^,  for  symmetry, 

P1P2  •  PiPi  =  -Pi-  PiPiPi  +  i>2  •  PzPiPi 
=      Pz-PaPxP-i-Pa-Pi 

The  last  expression  is  obtained  by  interchanging  the  suf- 
fixes 1  and  2  with  3  and  4.  If  in  (101)  and  (102)  we  put  the 
reference  points  e^,  Cj,  e^  for  the  p's,  the  equations  become  iden- 
tical, viz., 

p  =  e^-p\eQ  +  e^'P\e^-\-e.2-p\e^ (105) 

Similarly,  (103)  becomes 

p  =  eo-p|eo-l-ko-J3eo  =  eo-2>|^o  +  eoi)-!e„,  .     .     .     (106) 

a  form  analogous  to  eq.  (99),  expressing  p  in  terms  of  its  pro- 
jections on  60  and  6162- 

48.  The  operations  of  the  last  article  would  have  been  pre- 
cisely the  same  if  lines  (point-vectors)  had  been  used  through- 
out instead  of  points.  Hence,  substituting  i's  for  ja's  in  eqs. 
(101)  to  (106),  we  have 

L  =  -^—\_L,-LL,L,  +  L,'LL,L,  +  L,.LL,L,l   .     (107) 

L  =  -^—  \_\L,L, .  L\L,+\L,L,  •  L\L,  +  \L,L,  •  L\L.^,     (108) 

L0L1L2  L^L^^o 

^L,-L,UL.,-L,-L,L,L„ (110) 

L  =  eie.2- Le^i-\-e.2e(t- Lex-\- Bifii- Le.,, (HI) 

L  =  e^e.^  •  ZiCu  +  ey  •  LeyB^  =  eie^  •  Xjeie,  +{^\'^>  •  -i^)  •  \^\€->-     (112) 


48 


DIRECTIONAL   CALCULUS. 


[Art.  40. 


In  equations  (107),  (108),  and  (111)  L  appears  as  equal  to 
the  sum  of  its  projections  on  three  given  lines ;  in  (109)  and 
(112)  it  appears  as  the  sum  of  its  projections  on  a  line  and 

point.     The  projection  on  the  point  in  (109),  viz.      ^    ^ ' -, 

is  a  certain  portion  of  the  line  joining  tlie  common  point  of  i. 
and  L2  Avith  the  common  point  of  L  and  Lq,  the  reciprocal  idea 
to  that  of  the  projection  of  j?  onpiPa  in  (103). 

49.   In  eq.  (104)  put  p^Pi  =  \qi,  and  we  have 

In  (110)  put  L^L^  =  I  Jf],  and  we  have 

Aig  •  1^1  =  -  A  •  Ai^i  +  A>  •  Al^i-      •     •     •     (114) 

Note  that  qi  is  a  point,  the  anti-pole  of  P3p^,  and  M^  is  a  line, 
the  anti-polar  of  the  point  AA- 

Again,  in  (110)  put  AA  =  i'2j  A=|5'i>  A  =1^2;  then,  since 
Z3A  =  l9i  •  lg'2  =  |9i92,  we  have 

P2|^l92=|9l-i>2fe-  192-^1, (115) 

and  similarly  from  (104), 

Al  Ji/1^2  =  I-^i  •  A|^2  -  |i>/2  •  Al^i (116) 

Multiply  (115)  and  (116)  respectively  by  pi  and  A)  01*  (11<^) 
and  (114)  respectively  into*  \q2  and  \M2,  and  we  have 


PiPmiQi  = 


pm  pm 

P2\gi  P2\q2\ 


Al-^i    Al-^a 


(117) 


(118) 


If  qi=2h  and  go  =  ^25  -^1  =  A  and  Mo  =  L2,  (117)  and  (118) 
become 

PiP2\PxP2    =(PiP2)-  =PrP2--{Pi\P2y    \  .^^g. 

LiL,\LiL,  =  {L,L,y-  =  L^-W  -  (ii|  A)' )"     '     ^       ' 


*  A  multiplied  hy  B  means  BA  ;  A  multiplied  hUo  B  means  AB. 


Chap.  II.] 


MULTIPLICATION. 


49 


Put  ^2  iov 2)  in  (102),  and  multiply  hy  poPiPs'  goQi;  then, 
P0P1P2  ■  Qo(M2  =  QoQi\PiP2  ■  q-2iPo  +  Qoqi\P2Po  •  q2\Pi+  qoQilPoPi  •  QiIPs 


=  i>0|92 


i^ii^o  pm 

i>2|?0     P2I9I 


+Pi\q2 

+  P2\Q2 


P2\Qo  P2\9i 

PoIQo  Po\qi 

PoIQo  Poki 

pMo  Pi\qi 


(120) 


Po\qo  Pom  Pom 

=  Pi\qo  pMi  Pim 

i>2l9o  P2\qi  P2m 

Of  course  L's  and  M's  may  be  written  in  (120)  for^^'s  and 
q^s ;  i.e.  lines  may  be  substituted  for  points. 

Finally,  a  point  may  be  expressed  in  terms  of  two  points  in 
plane  space  as  follows.     Write 

p  =  xpi  +  yp.,, 
and  multiply  into  Pi\piP2  and  P2IP1P2  successively. 
•••  Plh\PiP2  =  yPiP2\PiP2  =  -  ViPiPi)-, 
PP2\PiP2  =  x(PiP.i)-; 
1 


whence       j^  = 


{Pi  •PP2\PiP2  -P2  -PPilPdh)- 


(121) 


(P1P2)- 

Note  that,  since  all  these  point  equations  are  homogeneous  in 
all  the  ^Joints  involved,  these  points  may  have  any  iveights  we 
please. 

50.  It  can  be  easily  seen,  from  the  geometric  interpretation 
given  to  combinatory  products,  that  the  equation 

AB  =  AC 

does  not  imj^ly  that  B=  C;  or,  in  other  words,  the  quotient  of 
A(B—  C)  divided  by  A  is  not,  in  general,  B—C.  Thus,  in 
plane  space 

pZ,  =  pL.> 

simply  means  that  the  two  quantities  are  the  same  in  magni- 
tude and  sign,  and  Li  and  L^  may  have  an  infinite  number  of 
relative  positions  and  lengths.     The  algebraic  reason  for  this 


60  DIRECTIONAL  CALCULUS.  [Art.  51. 

is  that  a  product  can  be  zero  without  either  factor  vanishing, 
so  that  division  is  indeterminate.     Thus, 

^'  =  L,  +  ix,2h-hx,p,)p, (122) 

because,  on  multiplying  both  sides  into  p,  the  equation  becomes 
an  identity.  As  the  subject  of  division  has  no  great  impor- 
tance in  the  developments  or  applications  proi^osed  in  this 
work,  it  will  not  be  further  discussed.* 

51.  ExEBCisEs. —  (1)  To  show  that 

tiTca  ±  caTci  and  Po(PiTpop2  ±2hT2Mh) 
are  the  respective  bisectors  of  the  angles  between  the  vectors 
ci  and  €2  a-'^d  the  point-vectors  poPi  and  Po2^2}  the  upper  signs 
corresponding  to  the  internal  bisectors  and  the  lower  to  the 
external. 

The  sum  and  difference  of  two  equal  vectors,  being  ||  to  the 
two  diagonals  of  a  rhombus,  evidently  bisect  the  two  angles 
between  the  vectors ;  hence 

Bisector  =  Uc^  ±Ue.  =  -^  ±-^  =  —^  {e.Te^  ±  €,Te,), 

which  is  the  first  expression  above  except  as  to  length.  The 
point  expression  is  found  in  the  same  way. 

(2)  A  parallel  to  a  side  of  a  triangle  cuts  the  other  sides 
proportionally. 

Let  ci  and  cg  be  ||  and  equal  to  two  of 
the  sides ;  then  t^  —  ci  is  ||  and  equal  to 
the  other  side.  Let  jh  —  ^o  =  ^i«i  aJid 
^>2  —  Co  =  ^2^2-  Then,  by  given  condi- 
tions, iCgCa  —  ^i«i  =  *i(E2  —  «i)  •  Multiply 
by  cj  and  cj  successively,  and  we  have 
ojjCiCa  =  WC1C2,  and  o^iCic,  =  ncjCg.     .•.  0Ci  =  x.,  =  n.  q.e.d. 

*  A  treatment  of  the  matter  will  be  found  in  the  fourth  chapter  of  the 
Ausdehnungslehre  of  1844,  and  a  more  extended  one  in  an  article  by  the 
Author  in  No.  1,  Vol.  IV.  of  the  "Annals  of  Mathematics,"  published  at 
the  University  of  Virginia. 


Chap.  II.] 


MULTIPLICATION. 


61 


(3)  The  bisectrix  of  an  angle  of  a  triangle  divides  the  oppo- 
site side  into  segments  proportional  to  the  adjacent  sides. 

With  the  figure  of  the  last  proposition  the  bisectrix  of  the 
angle  at  Cq  is  eQ{eiTe.2  ±  fioTei),  which  gives  the  proof  imme- 
diately, for  the  point  eiTejiesTci  is  on  the  line  6163  ^■t  dis- 
tances from  these  two  points  inversely  as  the  weights,  i.e. 
directly  as  Tt/  and  Te.,,  and  between  them  or  outside  according 
as  we  use  the  upper  or  lower  sign. 

(4)  If  a,  h,  c  are  the  three  sides  of  a  triangle,  to  show  that 

a^  =  6- -|- c- —  2  &c  cos  <  ,  • 
With  the  figure  above  let 

T{e,-e,)=a,     Tei  =  c,     Te,=  b. 

Then,  a'  =  T\e,  -  e,)  =  (e,  -  e,)'-  =  e,^  +  e,^  -  2  e,\e, 

c 
=  b'  -{-  c  —  2  be  cos  <  ,  • 

(5)  Find  the  condition  that  lines  through  the  three  vertices 
of  a  triangle  shall  have  a  common  point. 

By  the  figure  the  condition  is 

^olh  •  eah  '  62P2  =  0. 
Let  2)o  =  wiofii  +  ^0^25 

Pi  =  ^162  +  h%  e 

Ih  =  ^2^0  +  ^12^1 ; 
then, 

^oPi,  •  ei2h  •  e2i>2  =  e^im^ei  -\-  n^e^)  •  ei{nie^_  -f-  l^e^)  .  e.^Q^^  -f  m^ei) 
=  wioniZg  —  n^itn.2  =  0. 

This  is  equivalent  to 

Pii^-i  ■  IhSi)  •  Ih^i  —  ei2)o '  e^ih  •  60^2  =  0. 

(6)  Find  the  condition  that  three  points  on  the  respective 
sides  of  a  triangle  shall  be  collinear.     This  case  is  the  recip- 


52  DIKECTIONAL  CALCULUS.  [Art.  51. 

rocal  of  the  preceding.    Let  the  points  be  as  in  Ex.  (o) .    Then 
the  condition  is 

=  moniZ2+7i(,Zim2, 
which  is  equivalent  to 

P(fi2  •  Pi^o  ■  P^i  +  ^iPo  •  ejjPi  •  eoP2  =  0. 

(7)  Show,  by  Ex.  (5),  that  the  following  sets  of  lines  in  a 
triangle  have  a  common  point. 

1st.  Lines  through  the  vertices  and  the  middle  points  of  the 
opposite  sides. 

2d.  A  line  through  one  vertex  and  the  middle  of  the  oppo- 
site side,  and  two  lines  through  the  other  vertices  ||  to  the 
sides  opposite  to  them. 

3d.  The  bisectors  of  the  angles  ;  all  internal,  or  one  internal 
and  two  external. 

4th.  The  perpendiculars  from  the  vertices  on  the  opposite 
sides. 
5th.  The  perpendiculars  to  the  sides  at  their  middle  points. 

(8)  Show,  by  Ex.  (6),  that  the  points  where  the  bisectors 
of  the  angles  of  a  triangle  cut  the  opposite  sides  are  collinear, 
if  two  of  them  are  internal  and  one  external,  or  if  all  are 
external. 

(9)  From  the  values  of  a:^,  a^  iCg,  given  just  before  eq.  (101), 
determine  the  effect  upon  the  position  of  p  of  giving  a  negative 
value  to  one  or  more  of  these  coefficients. 

(10)  If  in  the  result  of  Ex.  (6)  lines  be  substituted  for 
points,  —  say  (Lq,  Li,  L^  for  (eo,  e^  e^)  and  {Lq,  Li,  L^)  for 
{Pf»  P\i  Pi))  —  interpret  the  resulting  equation. 

(11)  If  a  quadrilateral  be  divided  by  a  right  line  into  two 
quadrilaterals,  show  that  the  common  points  of  the  three  pairs 
of  diagonals  are  collinear. 


Chap.  II.] 


MULTIPLICATION. 


53 


(12)  By  substituting  lines  for  points,  in  the  equation  of 
condition  of  the  last  exercise,  derive  the  reciprocal  proposition. 

(13)  If  two  triangles  are  so  situated  that  the  lines  joining 
their  vertices  two  by  two  meet  in  a  point,  then  will  their  cor- 
responding sides  meet  each  other  in  three  points  lying  in  one 
right  line. 

(14)  Show  that,  if  a  line  L  cut  the  six  lines  that  can  be 
drawn  through  four  points  e^  62,  63,  64  in  the  six  points 

Pi,  Pi,  Ps,  Pi,  P2,  Ps, 

as  in  the  figure ;  then  the  relation 

.     Tp,po  •  Tp,p,' .  Tp2%'  =  Tp,%' .  TpM  '  TjhPs 

holds.     These  points  are  said  to  be  in 
involution. 

(15)  By  substituting  lines  for  points, 
and  a  point  p  for  L,  obtain  the  recipro- 
cal theorem,  and  interpret  it. 

(16)  Let  pi,  P2,  Ps,  Pi  be  four  fixed 
points,  and  let  p2  and  p^'  vary  subject 
to  the  conditions 

PiPM'  =P2PiPi  =PzP&Pi  =  0 ; 
find  the  locus  of  p,  the  common  point 
of  PiPi   and  P2PZ. 

If  PiP2Pz  =  ^,  show,  by  eq.  (104),  that  the  locus  becomes 
two  straight  lines,  one  of  which  passes  through  p^. 

We  have  at  once  P2  ^pPi-p^P*  a-^d  Pz  =PP2  •  PsP* ;  whence, 
by  substitution  in  above  condition,  we  have 

PiiPPz'PiPi) (PP2  -PzPi)  =  0, 

the  equation  of  the  locus,  which,  being  of  the  second  degree 
inp,  represents  a  conic.  On  applying  (104),  this  will  separate 
into  two  factors  of  the  first  degree  in  p,  if  P1P2PS  =  0. 


64  DIRECTIONAL   CALCULUS.  [Art.  5L 

(17)  Interpret  the  reciprocal  results  obtained  by  putting 
Z/'s  for  p's. 

(18)  If  Ci,  €.2,  63,  e^  are  four  coplanar  points,  and  e^  and  eg  are 
the  common  points  of  6162  and  6364,  and  of  6461  and  6963  respec- 
tively, show  that  the  middle  points  of  6163,  6^64,  and  e^Sf.  are 
coUinear, 

(19)  Lines  through  the  vertices  of  any  triangle  and  the 
corresponding  vertices  of  its  complementary  triangle  meet  in 
a  point ;  and,  reciprocally,  the  corresponding  sides  cut  each 
other  in  three  coUinear  points. 

(20)  A  triangle  whose  sides  are  of  constant  length  moves 

so  that  two  of  its  vertices  remain  on  two 

fixed  straight  lines  :  find  the  locus  of  the 

other  vertex.     Let  ^o^i  and  eo«2  be  the  two 

fixed  lines,  and  PiP-iP  the  triangle.    Also 

let  pi  —  Co  =  xei  and  p2  —  €0  =  y^^ ;   then 

J92  —  i>i  =  2/£2  —  Xf-D  and  we  have  the  con-  ^0 

dition  rr/  \ 

T(ye2  -  xe{)  =  c. 

Let  pe  be  A.  to  P1P2,   Tpie  =  mc,   Tep  =  nc ;  then, 

p  —  eQ=p  =  xti-\-m  (2/62  —  a^ci)  +  n\  {yt.^  —  xt^) . 

This  equation  in  p  and  the  scalar  variables  x  and  y,  with  the 
condition  above,  which  is  really  of  the  second  degree  in  x  and 
y,  is  that  of  a  conic  section,  which  must  evidently  be  an  ellipse. 
The  student  should  eliminate  x  and  y  by  multiplying  succes- 
sively by  cj  and  €2,  thus  obtaining  a  scalar  equation  in  p  of  the 
second  degree. 

(21)  Show  that  the  expression  ppiLipJL^PzP',  interpreted 
according  to  Art.  16,  (c),  is  identically  equal  to 

{pPi-Li)P2{Li-mx>'), 

and  from  this  that  it  is  also  equal  to  —  l^'lhl^iPzLiPiP 


Chap.  II.] 


MULTIPLICATION. 


55 


By  eq.  (104), 
pPiLiPoLzPsP'  =l)PiLiP>(Ps'P'L2  —p'  -ihLo) 

=  (ppi '  A  -P^lh)  -P'L.,  -  (p2h  ■  Li-pop')  -p^Li 
=  (P1P2P3  -pLi  -pPiPi  -ihLi)  -p'L^ 

-  {P1P2P'  -pLi  -pp.p'  -p-^L-^)  'p.^L., 
=  (Pi  -pLi  -p  'PiLi)p2(p3-p'L2  -p'  -PzLi) 
=  (PPi-L,)2h{L.2-p3P')- 
The  second  part  is  left  to  the  student. 

(22)  If,  as  in  eq.  (86), 


show  that 


'0 


T\p  =  TL  =  y/{lo-  hYci  +  (Zo  -  ^2)  V  -  2(/o  -  h)  {k  -  k)^^^ 
in  which  tj  =  e^  —  e^  and  (.2  =  6.2  —  e^. 

(23)  If  Li=Piih\  L2=2hl^2)  J^s^PsPs'}  then  show  that 

L1L2L3  = 


2hL2  PiL^l 
P1L2  Pi'Lsl 


2hLs  PiLi 

P2'Ls    P2'Li 


2h  Li  2h  L2 
p^Li  p^Li 


(24)  By  eq.  (119)  prove  that 
{xi  +  Vi  +  ^i)  {X2'  +  yi  +  zi)  =  {x^X2  +  2/,2/ii  +  z^Z2y 


+ 


2/1  ^1 
2/2  % 


+ 


1 2^2    ^2 


+ 


X2  2/2 


(25)  Show  by  eq.  (120)  how  the  product  of  two  determi- 
nants of  the  third  order  may  be  expressed  as  a  determinant  of 
the  same  order. 


(26)  If  Zi  =  2o(?|e)  and  L2=%){m\e),  show  that  when 

'2      '0 


mo  mi 


+ 


Im,  m2 


+ 


=  0, 


then  Li  and  L^  are  parallel. 


56 


DIRECTIONAL   CALCULUS. 


[Art.  52. 


Stereometric  Products. 

52.  Three-dimensional  space  is  the  locus  of  all  points 
dependent  on  four  fixed  points.  Let  these  four  reference 
points  be  Cq,  Cj,  e^,  e.^,  so  situated  relatively  to  each  other  that 
60616263  =  1,  always ;  i.e.  the  unit  of  volume  is  six  times  the 
volume  of  the  reference  tetraedron.  Let  four  points  be 
taken,  viz. : 

2) I  =  Sofce,  2h  =  So^e,  ps  =  2om6,  2h  =  So^e ; 

then, 


P1P2  =  Li : 


6061  + 


+ 


'2    ^3 


^0^2     I 


62^3  + 


'0  h 

/to     AJj  I 

'3    'l 


6,6, 


1(123) 


"'0  "'I  "'2  "'S 
to    ti    12    '3 


[60,   61,   62?   63]- 


The  first  result  will  be  obtained  by  actual  multiplication  of 
the  values  of  pi  and  p2,  and  the  second  result  is  simply  an 
abbreviated  way  of  writing  the  other  as  in  eq.  (55).  It  ap- 
pears thus  that  any  point-vector  in  space  is  expressible  in 
terms  of  the  six  edges  of  the  reference  tetraedron. 

Again, 


P1P2P3  =Pi  = 


Kq 

k, 

/to 

kg 

k 

h 

k 

h 

mo 

mi 

m^ 

rris 

[60,  ei,  62,  63] 


|eo 

ki 

1^2 

h 

kf) 

^'1 

/Co 

kg 

k 

h 

h 

k 

mo 

m, 

m, 

vris 

y, 


(124) 


in  which  the  third  member  means  the  sum  of  the  four  third- 
order  determinants  that  can  be  formed  of  the  columns  taken 
three  at  a  time,  each  multiplied  into  its  corresponding  triple 
product  of  the  reference  points,  with  the  same  order  of  suf- 
fixes. In  the  fourth  member  160=616263,  |6i=— 606360,  162=636061, 
and  [63  =  —  606162.     Thus  any  point-plane-vector  is  expressible 


Chap.  II.]  MTTLTIPLICATION.  57 

in  terms  of  the  four  faces  of  the  reference  tetraedron.      Of 
course  the  product  of  three  points  is  not  scalar  in  solid  space. 
Finally, 

A'o  ki  k.2  kg 

P1P2I  ^i  *~  ffio  nil  m2ms^ 

Hq  ill  n.2  Hg 

because  60^16263  =  1 ;   thus  the  product  of  four  points  is  scalar, 
as  was  shown  in  Art.  22. 

As  an  exercise  let  the  student  find  the  condition  that  the 
plane  P4,  in  (124),  shall  pass  through  the  mean  of  the  refer- 
ence points. 

53.  Since,  by  Art.  17,  the  continued  product  of  four  points 
obeys  the  associative  law,  we  have 

PlPsPsPi  =PlP2Pz  -Pi  =  P*Pi  =PlP2-PsPi  =  LiLi 

=  -p,P,  =  LoLi (126) 

Thus  the  product  of  a  point  and  plane  is  now-commutative, 
while  that  of  two  lines  in  solid  space  is  commutative.  The 
stereometric  product  of  two  lines  is  according  to  Art.  16,  (a), 
while  the  planimetric  product  is  according  to  Art.  16,  (6). 

54.  Product  of  a  line  and  a  plane.  Let  L  be  the  line,  and  P 
the  plane,  and  let  po  be  the  point  where  the  line  pierces  the 
plane.    Take  pi,  p2,  Ps,  so  that 

L=poPi  and  P^lhPsPa- 

.-.  LP  =  poPi'Polh2h=PoPiP2p3'Po (127) 

This  is  in  accordance  with  Art.  16,  (6),  and  the  model  form 
of  eq.  (59) .     Also, 

PL  =  PQP2P3 '  PoPi  =  PoPoPsPi  •  Po  =  P0P1P2P3  •Po  =  LP;  (128) 

so  that  this  product  is  commutative,  like  pL. 

If  L  is  parallel  to  P,  po  is  at  00,  and,  replacing  it  by  e,  we 
have  for  this  case 

PL  =  LP=€2h'^P2Ps  =  ^iP2P3'^-     ....     (129) 


58  DIRECTIONAL   CALCULUS.  [Art.  55. 

55.  Product  of  two  planes.  Let  them  be  Pj  and  P.y,  and  let 
L  be  their  common  line,  while  Pi  and  2>2  are  so  taken  that 

Pi  =  Lpi  and  Pg  =  Z^jg- 
Then,  P^Po^=  Lp,- Lp.,  =  Lp^p.,- L  | 

P,P,  =  ^>,-^>,=:^Ijp,p,.L  =  -P,Pj'  ^  ^ 
so  that  the  product  of  two  planes,  like  that  of  two  points,  is 
non-commutative. 

If  Pi  and  P2  are  parallel,  i  is  at  oo  and  becomes  a  plane- 
vector  ;  call  it  -q,  and  substitute  in  (130) ;  then  we  have  for 
the  product  of  two  planes,  having  a  common  line  >;  at  00, 

PiP2=^Vlh-vP2  =  vPiP2-v  =  -P2Pi-     •     .     (131) 

56.  Product  of  three  planes.  Let  jh  be  the  common  point  of 
Pi,  P2,  and  Pg,  and  take  pi,  p^,  jh  on  the  common  lines  of  these 
planes,  so  that 

,,  Pl=PoP2P3,      P2=PoP32h,      P3=PoPlP2; 

then, 

P1P2P3  —  P0P2P3  •PoP3Pi-PoPiP2  =  -  023  .  013 .  012  ] 

=  -  023  •  0132  .  01  =  0123  .  023  .  01  I      (132) 

=  0123  .  0231 .  0  =  {poPiPoPsY  'Po  J 

In  this  equation  we  have  used  0  for  p^,  1  for  pu  etc.,  for  con- 
venience. This  we  may  frequently  do  when  no  ambiguity 
will  result.  In  eq.  (132)  we  have  Avorked  according  to  Art. 
16,  (c),  by  which  PiP2P3  =  Pj  •  P2P3;  but  if  we  had  combined 
Pi  and  P2  first,  and  the  result  with  Pg,  we  should  have  obtained 
the  same  result.  Hence  planes  obey  the  same  laws  *  of  multi- 
plication as  points,  in  solid  space. 

*  We  have  here  assumed  the  distributive  law  to  hold,  as,  in  fact,  it 
does,  for  all  products,  progressive,  regressive,  or  mixed  ;  but  it  is  easy  to 
prove  the  law  for  planes  or  lines,  assuming  it  to  be  true  for  points.  Thus, 
taking  the  planes  as  above, 

P,P,  +  P1P3  =  023  •  031  +  023 .  012  =  0123  •  (03  -  02),  because  0123  is  scalar, 
=  0123  .  0(3  -  2)  =  01  (2  -  3)3  •  0(3  -  2) 
=  0(3  -  2)13  •  0(3  -  2)  =  0(3  -  2)  1  .  0(3  -  2)3 
=  023  .  01(2  -  3)  =  023  •  (031  +  012)  =  Pi(P,  +  P,). 


Chap.  II.]  MULTIPLICATION.  59 

If  the  three  planes  are  parallel  to  one  right  line,  the  common 
point  is  at  oo,  and  c  may  be  substituted  tov  po  in  (132). 

57.  Product  of  four  planes.  Let  pi,  p.2,  Ps,  p^  be  the  four 
common  points  of  four  planes  P^  P.2,  Pg,  P4  taken  three  by 
three,  and  take  four  coefficients  7ii,  •••714,  so  that  Pi  =  "iPai^ai^^i 
etc. ;  then 

P1P2P3P4  =  n,n,n^n^  •  234  •  341  •  412  •  123  ) 

=  n^n^n^niipiP^PsPiY  ) 

Mixed  products  are  to  be  interpreted  according  to  Art.  16, 
(c).     Thus, 

L,P,L,P,L,  =  L,\P,IL,{PM^\ 

has  this  meaning.  PoL^  is  a  point;  this,  multiplied  by  i,? 
gives  a  plane ;  this,  by  P^,  a  line ;  and  this,  by  Li,  a  scalar 
quantity, 

58.  Products  of  plane-vectors.  Let  -qi  and  172  be  two  plane- 
vectors  (lines  at  00  ),  and  let  c  be  parallel  to  each  of  them, 
while  cj  and  €3  are  so  taken  that  -qx  =  etj  and  r^o  =  ^^2  j  then, 

rjirj.2  =  eci  •  £€2  =  etiCo  •  e (1^^) 

This  result  may  be  obtained  directly  from  eq.  (58)  by 
regarding  the  points  and  lines  of  that  equation  as  all  at  00, 
and  therefore  necessarily  in  the  plane  at  00. 

The  product  of  two  plane-vectors  appears  as  a  vector  parallel 
to  each  of  them,  multiplied  by  a  scalar  quantity.  We  have 
at  once 

V'2Vi  =  ~  V1V2 (135) 

Next  take  a  third  plane-vector  rjs,  and  let  ci  be  ||  to  772  and 
rjs,  €2  II  to  7/3  and  Tji,  C3  II  to  r/j  and  r]2,  while  the  tensors  of  cj,  etc., 
are  such  that  r]i  =  €2«3»  r]>  —  ^s'n  Va  —  '1^2  5   then 

'7l^2'?3=C2C3-C3ei'Cie2  =  («ie2f3)" (1^6) 

As  an  exercise  let  the  student  discuss  rjP,  the  product  of  a 
plane-vector  and  a  point-plane-vector. 


60  DIRECTIONAL  CALCULUS.  [Art.  50. 

59.  Equations  of  condition.     By  eq.  (48), 

pP=0 (137) 

makes  p  lie  on  P,  or  P  pass  through  p ; 

L,L2  =  0 (138) 

makes  the  two  lines  intersect ; 

LP  =  0 (139) 

makes  L  lie  in  P,  or  P  pass  through  L ; 

PiP2  =  0       (140) 

makes  the  two  planes  coincide ; 

PrP,P,=  0 (141) 

makes  the  three  planes  pass  through  a  common  line ;  for  P^Ps 
is  a  line,  say  L,  and,  by  (139),  PiL=0  makes  P^  pass  through  Z; 

PiPoPzPi  =  0 (142) 

makes  the  four  planes  pass  through  one  point,  for  P^P^P^  is 
some  point,  say  p,  and  pP^,  by  (137),  makes  P^  pass  through  p ; 

PiLP2=0 (143) 

makes   Pj   and  Pg  cut  L  at   the   same   point ;    for,   writing 
L  =  P3P4,  the  result  follows  from  (142)  ; 

^.^2  =  0 (144) 

makes  the  two  plane-vectors  parallel ; 

i7ii72'73  =  0 (145) 

makes  the  three  plane-vectors  all  parallel  to  one  straight  line. 
Equations  (138),  (140),  (141),  (142),  (144),  (145),  should 
be  compared  with  equations  (66),  (46),  •••  (50),  respectively. 

60.  Addition  of  planes  and  plane-vectors.  Let  Pj  and  Pg  be 
two  planes  intersecting  in  L,  and  let  pi  and  p^  be  so  taken  that 
Pi  =  Lpi  and  Pq  =  Lp2 ;  then 

Pi  +P2  =  L20,  +  Lp,  =  L(p,  +p,)  =  2Lp,      (146) 

in  which  p  is  the  mean  of  jj^  and  pa-     Thus  the  sum  is  that 


Chap.  II.]  MULTIPLICATION.  61 

diagonal  plane  of  the  parallelopiped,  of  which  two  adjacent 
faces  are  Pi  and  P^,  which  passes  through  L\  the  parallelo- 
grams Pi  and  Pa  heing  so  placed  as  to  have  a  common  side  L. 

If  the  t'  vo  planes  are  parallel,  let  iy  be  a  plane-vector  parallel 
to  eacli  of  them,  i.e.  their  common  line  at  oo,  and  let  pi  and  p^ 
be  points  of  the  respective  planes ;  then  we  may  write 

whence  P\  + P-i  =  '^\riPi  +  n2-nP2  =  v{'*hPi  +  niP2)\     ,^.„. 

=  (^1  +  Wa)]?!?  i 

If  111  -f  ^2  =  0,  then 

P,  +  P,  =  n,{p,-p,)y^, (148) 

so  that  the  sum,  in  this  case,  becomes  a  volume,  and  is  scalar. 
Cf.  eq.  (74). 

Take  Pj,  Pg,  Pg,  j9o>  i?i>  Pi^  Pi  as  in  Art.  56 ;  then 

Pi-\-Pi+Ps=p^{P2Pz+PzPi+PiP2)  =Po(P2-Pi)  {P3-Pi)-   (149) 

Thus  the  sum  is  a  plane  through  the  common  point  parallel 
to  the  plane  PiP^Pz- 

li Pq  is  at  CO,  call  it  c ;  then  each  plane  is  ||  to  c,  and  the  sum 
becomes  the  product  of  three  vectors,  and  therefore  Scalar. 

If  Pi-fP2  =  0,  or  Pi  =  -P2, (150) 

the  two  planes  are  coincident. 

If  Pi-fP2-f-P3=0, (151) 

the  three  planes  pass  through  one  right  line,  as  appears  by- 
comparison  with  eq.  (146) . 

Similarly,    Pi  + P^  + P^+ P^  =  0 (152) 

causes  the  four  planes  to  pass  through  a  common  point,  as 
appears  from  eq.  (149). 

Take  rji,  772,  e,  ci,  ^2  as  in  Art.  58 ;  then 

i7i  +  >72  =  «i  +  «2  =  c(ei  +  c2),       ....     (153) 

so  that  the  sum  is  a  plane-vector  parallel  to  c. 


62  DIRECTIONAL   CALCULUS.  [Akt.  01. 

61.  Addition  of  point-vectors,  or  lines.  Take  n  point-vectors 
Pi^i}  P2^2>  '"Pn^nf  ^^^  Call  tlicir  sum  S ;  then 

- .    (lo4) 
=  eoSc+2(i)-eo)£  ) 

It  appears  that  S  is,  in  general,  composed  of  two  parts,  of 
which  one  is  a  ^wmf-vector,  and  the  other  a  pZane-vector.  If 
this  plane-vector  is  parallel  to  the  point-vector,  i.e.  capable  of 
expression  as  the  product  of  some  vector  a  into  Se,  then  their 
sum  can  be  expressed  as  a  point-vector  only ;  for  we  have,  in 
this  case, 

S  =  eoSe  +  aSc  =  (^0  +  a)  Se, 

a  point-vector  of  the  same  length  as  Co^e,  ||  to  it,  and  distant 
from  it  by  the  amount  Ta  sin  <    '• 

S  being  composed  of  two  parts  which  cannot  be  equal  to 
each  other,  if  we  have  the  equation  S  =  0,  it  can  only  be  satis- 
fied by  making  each  part  separately  zero,  so  that  S  =  0  implies 
Sc  =  0  and  2(p  — eo)c  =  0.  The  quantity /S"  maybe  called  a 
screiv,*  and  we  shall  hereafter  consider  some  of  its  properties. 

62.  TJie  complement  in  three-dimensional  space.  Following 
the  definitions  of  Art.  38,  we  have  for  a  unit  normal  vector 
system  ii,  tju  hf 

[ij    =    Ijtg,  |(,2l3    =  ||(,i    =    Ij    V 

|l2    =    lotj,         [iglj    =||l2   =    l^     f (15^) 

Let  cj  =  Zjii  +  ^212  +  Z3I3 

and  C2  =  Willi  +  m^i^  -f-  wijia ; 

then  |ci = Z,t2i.3 + Wi + hhh  =    (Jih — ^2ti)(?it3 —kh),    (156) 


*  See  "The  Theory  of  Screws,"  by  R.  S.  Ball,  Dublin,  Hodges,  Foster 
&  Co.  ;  and  also  a  paper  by  the  author  on  "The  Directional  Theory  of 
Screws,"  Annals  of  Mathematics,  Vol.  IV.,  No.  5. 


Chap.  II.] 


MULTIPLIC  ATION. 


63 


SO  that  |ei  is  a.  plane-vector.  The  third  member  of  (156)  is  the 
product  of  two  vectors ;  the  first,  l^i^  —  Z^'u  is  easily  seen,  by 
the  figure,  to  be  X  to  Zjii  +  ^212*  the  projection  of  cj  on  the  plane 


tii2,  and  hence  _L  also  to  cj,  because  JL  to  the  plane  that  projects 
cj  on  ijia ;  similarly,  I^l^  —  l^ii  is  X  to  Zjii  +  ^313,  hence  to  the  plane 
that  projects  ci  on  tgij,  and  therefore  to  ej  itself. 

Hence  [ej  is  a  plane- vector  perpendicular  to  c^.  Since  ||ci  =  cj, 
it  follows  that  the  converse  is  true ;  that  is,  the  complement  of 
a  plane-vector  is  a  line-vector  perpendicular  to  it. 

It  is  evident  from  the  figure  that  ci  is  a  diagonal  of  the 
rectangular  parallelepiped  whose  edges  are  li,  l^,  k  in  length ; 
hence, 

(157) 


Ter=^h'  +  li  +  l,\       .      .     . 
Multiply  (156)  by  cj ;  therefore 

cik  =  ci^  =  h-  +  k'  +  I3'  =  T\, 


.     (158) 


so  that,  as  in  plane  space,  the  co-square  of  a  vector  is  equal  to 
the  square  of  its  tensor.  The  product  cijcj  is  that  of  the  vector 
cj  into  a  ±  plane-vector,  as  has  just  been  shown ;  it  is  therefore 


64 


DIRECTIONAL   CALCULUSL 


[Art,  63. 


a  volume  which  is  equivalent  to  Tcj  times  the  area  of  Icj ;  hence, 
by  (158),  the  area  of  jti  is  numerically  equal  to  Ttj,  or 

T\e.  =  Tc,.- (159) 

Thus  the  complement  of  a  vector  in  solid  space  is  a  perpendic- 
ular plane-sector  having  the  same  tensor. 

We  have  y 

£j|c2=(Zili  +  ?2l2+?30(^l'2t3+W'2l3ll+Wl3'l'2))      f^ra\f        ' 

=  Ijmi  +  l^m.^  +  Zgmg  =  cgK  > 

Now  ci|c2,  being  the  product  of  the  vector  c, 
into  the  plane-vector  jcj,  is  equivalent  to 


that  is, 


Tci .  T  |£2 .  sin  <  1^2  ^  Tei  Te^  cos  <  ^' : 
^1  ^1 


Ci|c2  =  C2k  =  h'n^l  +  ^2*'''2  +  ^3^13  =  TeiT€2  COS  < 


^2 


(160) 


If  cj  and  C2  were  unit  vectors,  Zj,  L,  Is,  m^,  m2,  m^  would  be 
direction  cosines,  and  thus  (160)  gives  a  proof  of  the  formula 
for  the  cosine  of  the  angle  between  two  lines  in  terms  of  the 
direction  cosines  of  the  lines. 

By  (160)  the  condition  that  ci  and  eg  shall  be  at  right  angles 
is 


cil£2  =  0 ■. 

Let  rji  =  \ei  and  r]2=  |e2  5  then 


V2 


'7ll'/2=  kl  •  «2  =  C2|«l  =  eik2=  r£irC2C0S<  ^2  =  rniTw2C0S<       . 

and  rji\r]2  =  0 

is  the  condition  of  perpendicularity  of  two  plane-vectors 


(161) 

(162) 
(163) 


63.  Complement  in  a  point  system  in  three-dimensional  space. 
Let  eo,  fii,  e.2,  e^  be  four  unit  reference  points,  so  taken  that  the 
product  60^1^263  =  1 ;  then 

|^0^1  =  ^2^3>  ||^1^|^2^3^=^0*?1> 

I  ^0^2  ^=  ^3^1»         1 1  ^0^2  ^^  I  ^3^1  ^^  ^0^2> 
|^0^3  =  ^l62>         ||^0^3=|^1^2=^^0% 


1^0^=  ^1^2^3? 
1 61=  —  62^360, 
1^2^  6360^], 
163^=  — 60^1^2? 


ro —        riC2^3 —  — PQ, 

\ei=  —\e.2esen=  —ei, 

1^2^^=         I^S^O^l^^  — ^2) 

,k3=— 1^0^162  =-63, 


Chap.  II.]  MULTIPLICATION.  65 

Note  that  the  complement  of  the  complement  of  a  reference 
point  is  the  point  with  negative  sign,  but  that  the  complement 
of  the  complement  of  a  reference  line,  or  edge  of  the  reference 
tetraedron,  is  the  line  with  positive  sign.     We  have 

ko  •  l^l  •  1^2  =  —  61^263  •  626360  •  636061  =  (60616263)2  .  63  =  63  =  1  (606162), 

which  agrees  with  Art.  38,  (c). 

Let    pi  =  2oA;e,  and  p^  =  So^e ;  then 

1Pi  =  SoA^le  =  koeiezCs  +  etc.,  \ 

\Kl        Kq/\K2        Ko/\K^        KqJ  } 

SO  that  the  complement  of  any  point  is  a  point-plane-vector, 
or  plane,  and  any  plane  may  be  expressed  in  terms  of  the  four 
faces  of  the  reference  tetraedron. 

From  eq.  (123)  it  follows  that  the  complement  of  any  point- 
vector  or  line  is  another  point-vector.     Again, 

i>ili>2  =  2A;6| 2Ze  =  ZqWIo  -f  ?imi  -f  l^m^^  +  Zgrng  =  i>2|2>i.  •     •     (165) 

Let  Pi  =  |pi,  P-i  =  \pt ;  therefore, 

Pi|P2=IPi-|Ii>2=-IPi-P2=P2lPi=i>i[P2  =  P2lA.     .     (166) 

Let     ii  =  ^16061  -f-  ^,6062  -f  A;3eo63  -f  fci'e2e3  -f  A;2'63ei  -f  li-ie-fi^ 
and         L2  =  ^16061  + 1-26062  -f  etc. ; 
then       Li\L2=Jc,li+k2k+hk+kiV+hV+J<:A'=Lo\Li.    (167) 

Also   LiL2  =  kik'  +  k^2'  +  hk'  +  h'h  +  h%-\-ks%.     .     (168) 

If  L2  =  Li,  we  have 

L,'  =  2{kJc,'  +  kJc2'  +  kM (169) 

But  if  Li  is  a  point-vector,  its  square  must  be  zero,  and  as 
the  second  member  of  (169)  is  not  necessarily  zero,  it  follows 
that  Li  and  L2  are  not,  in  general,  point-vectors ;  in  fact,  they 
are  screws,  as  shown  in  Art.  61.  We  have  then  for  the  condi- 
tion that  Li  shall  be  a,  point-vector, 

kJCi'  +  k2k2'  +  hks'  =  0 (170) 

From  (165),  (166),  and  (167)  it  appears  that  a  co-product 
in  which  the  factors  on  opposite  sides  of  the  sign  are  of  the 


66  DIRECTIONAL  CALCULUS.  [Art.  64. 

same  order  is  commutative  about  that  sign,  and  always  scalar. 
If  the  factors  are  not  of  the  same  order,  this  is  not  the  case ; 
for  example, 

P\L  =  -';XP\L)  =  -\{\P-L)  =  -\{L\P).    .     .     (171) 

Proceeding  in  a  similar  manner  to  that  of  Art.  43,  it  may  be 
easily  shown  that  \p  is  the  anti-polar  plane  of  p,  with  reference 
to  an  ellipsoid  so  situated  that  each  vertex  of  the  reference 
tetraedron  is  the  anti-pole  of  the  opposite  face.  If  the  refer- 
ence tetraedron  is  regular,  and  tt  be  one  of  its  equal  edges,  the 
ellipsoid  becomes  a  sphere  whose  radius  is  easily  found  to  be 

a 

With  this  geometric  interpretation 

PiIP2  =  0 (172) 

causes  pi  to  be  in  the  anti-polar  plane  of  2h  with  reference  to 
the  reciprocating  ellipsoid,  and  vice  versd ; 

P^lP2  =  0 (173) 

causes  Pi  to  pass  through  the  anti-pole  of  P2,  and  vice  versd; 

A!^  =  0 (174) 

causes  Li  to  intersect  the  anti-polar  line  of  X,,  and  vice  versd; 

p\P=0 (175) 

makes  p  the  anti-pole  of  P. 

64.  All  the  quantities  we  have  to  deal  with  in  three-dimen- 
sional space  —  viz.  scalars,  points,  lines,  screws,  and  planes  — 
are  expressible  in  terms  of  fifteen  quantities,  which  are  all 
either  the  reference  points  or  products  of  them  of  different 
orders ;  they  are  the  four  reference  points ;  their  six  products, 
two  by  two,  i.e.  the  edges  of  the  reference  tetraedron ;  their 
four  products,  three  by  three,  or  the  faces  of  the  reference 
tetraedron ;  and  the  product  of  the  four,  which  is  numerical 
unity.  A  multiplication  table  can  be  easily  constructed  sim- 
ilar to  that  in  Art.  45.  Considered  as  an  algebra  it  appears 
that  this  system  is  Jifteen-fold. 


Chap.  II.]  MULTIPLICATION.  67 

65.  Projections.  We  have  the  same  fundamental  formula 
for  projection  as  in  Art.  46,  viz.  : 

(Projection  on  B  of  A,  directed  by  C)  =        ^   ,      .     (176) 

in  which  BG  is  scalar,  while  B  and  C  separately  are  not.  If 
we  substitute  in  the  equations  of  Art.  47  vectors  for  points, 
and  plane-vectors  for  point-vectors,  we  shall  obtain  a  set  of 
corresponding  formulae  for  a  vector  system  in  solid  space,  as 
follows : 

P  = (ei-p«2£3  +  «2-pe3ei  +  £3-p£ie2),      .      .      .      (177) 

ei«2f3 

«= (le2«3  •  p!«i  +  ks'i  •  ph -I- |ci«2  •  pI^s)?     •      •      (178) 

__  ^1  •  P^2^3     I     ^2^3  •  P^l  rl79^ 

flC2«3  C2C3fl 

These  are  derived  from  eqs.  (101),  (102),  and  (103),  and 
the  last  one  gives  p  in  terms  of  its  projections  on  ci  parallel  to 
C2C3,  and  on  cgCg  parallel  to  c^.  Also,  from  (104),  (105),  and 
(106),  we  have  ' 

C1C2  •  C3C4  =  —  Cl  •  e2<3«4  +  C2  •  e3C4ei  =  C3  •  C4«lC2  —  £4  •  ClC2«3J  (180) 

P  =  i-i- p\h  +  h- p\h  +  h'p\h! (181) 

=  h  •  p\h  +  i'l  •  p'l  =  1-1  •  p\i-i  -\-  i-iP  •  I'l (18-) 

If  7],  rji,  etc.,  are  plane-vectors,  we  have  from  the  equations 
of  Art.  48, 

V  — {vi'vn^Vi  +  Vi'vvsvi  +  vs-vviVi)}     ■    •    ■    (183) 

= ilviVi-V^Va  +  lviVa-vlvi  +  lvsVi'vlVs))      •     •      (1^4) 

'7i'72'73 

__-ni  ' -nyiiiji       ■qiqz-yrii^ (185) 

"~     ■»?i'72'?3  '?2'73'7l 

■niVi '  V3Vi=  —vi '  ViViVi-^Vi  •  V3n4Vi=V3  •  v*viV2—Vi  •  viv^fii,  (186) 

,y  =  Mg  .  lytl  -f  I3I1  .  ,yt2  +  1^12  .  T/lg, (187) 

=  l2*3  •  V'l  +  *1  •  '7^/3 (188) 


68 


DIRECTIONAL   CALCULUS. 


[Art.  66. 


Finally,  from  the  equations  of  Art.  49,  Ave  have 

ei£2  -Ici'  =  —  Ci  •  Csjei'  +  ^2  •  f^Mi,        •      • 

vhi'vi  =  \vi  •  vh2  —  W '  ViW)  •    • 


I 


CzKi     C2K2 
C1C2IC1C2  =  (fie2)-  =  eiV  —  (cl|«2)^ 


(189) 
(190) 
(191) 
(192) 

(193) 


(194) 
(195) 
(196) 


Cll^l       ei|C2       C]|C3 
ClC2«3  •  «l'«2'«3'  =    £21^1'     £21^2'     £2^3 

C3l«i'    ^sh'   ^ah' 
P  •  (ci«2)-  =  £1  •  pe2lei£2  —  C2  •  pCi]ciC2 
From  eq.  (192)  on,  the  plane-vector  equations  have  not  been 
written;   to  obtain  them  we  have   only  to   substitute   plane- 
vectors  for  vectors  in  eqs.  (193)-(196). 

66.   Projections  in  a  point  system.     Write 

P  =  X^Pii  +  XiPi  -f  052^2  4-  ^iPz 

and  multiply  successively  by  jhlhPs)  PiPsPn,  ^tc,  and  we  find 


Xi\  — 


PPxPiPi 


PPoJhPo 


etc. 


whence 
P 


P0P1P2P3  P1P2P3P0 

(PO  •  PP1P2P3  -  Pi  •  PP2PzPii  +  Ih  •  imihlh 


P0P1P2P3 

-Ps-PPoPiP2),    ....     (197) 

which  gives  p  in  terms  of  its  projections  on  any  four  points. 
Write  next 

p  =  XoPo  +  Xi2h  +  352^2^3  'lypol^u 

and  multiply  successively  by  p^p^Ps,  PiPsPm  and  PsPoPi ;  the 
values  of  a^  and  Xi  will  be  the  same  as  before,  but  that  of  X2 
1 


will  be  X, : 


P0P1P2P3 


;  hence 


P  = 


P0P1P2P3 


-{Po-PPlP2lh-lh  •PP2P32h+P2P3-PPoPl),      (198) 


Chap.  II.]  MULTIPLICATION.  69 

which  gives  /:>  as  the  sum  of  its  projections  on  p^^  pi,  and  the 
line  p^Ps-  Similarly,  we  have  for  the  expression  of  a  point  in 
terms  of  its  projections  on  any  point  j>i  and  any  plane  Pj, 

Pl^  1  J^lPl 

and  for  the  expression  in  terms  of  its  projections  on  any  two 
lines  Li  and  L.,, 

^,  =  L,.pL,^L,.pL, ^200) 

In  equations  (197) -(200)  the  projected  point  p  may  be  replaced 
by  a  screw  S,  or  a  plane  P.  We  may  also  write  in  (197)  and 
(198)  planes  for  points  throughout. 

Let       Pi^PoPsPm    P2=P3PoPi,   etc.; 

the  projection  of  |)  on  IPj  directed  by  l^^i  is  ! — '  'PiP^^  and  we 

may  write 

«  _  \P*>'P\P<'     ,     \Pl'P\Pl     ,     \P2'P\P2     ,     \PZ-P\P-^  .of^-,  X 

or,  taking  the  complement  of  both  sides, 

\P  =  P=:^;^^iP^-P\P^^-P^'P\P^+P-rP\P^-P^'P\P^)-  (202) 
PoPiPiPz 

Let  there  be  two  planes,  P  =  Pi2hP3  and  Q  =  qiq^s ;  then 

PQ  =  PiPtPz '  (h<M-6  =  m-m  +  y<Mi  +  zq^q^,  say. 
Multiply  both  sides  into  q^^h ;  then 

P1P2PZ  ■  qiq^qz  •  qiPi  =PiP2Pz'h  •  gi?2^3i>i  =  m'/M\Pi> 

and  X  =  Pqi. 

Similarly,  y  =  Pq.^  and  z  =  Pq;^. 
Hence, 

PQ  =  Pqi .  q^qs  +  Pq^  •  q^qi  +  Pq^  ■  q^qi 


(  7       -      (203) 

PiPz-PiQ  +P3P1  -p^Q,  +P1P2  'PsQ ) 

the  second  value  being  found  in  the  same  way  as  the  first. 
This  equation  expresses  the  common  line  of  P  and  Q  in  terms 
of  three  points  of  P  or  Q. 


70 


DIRECTIONAL   CALCULUS. 


[Art. 


The  terms  of  the  second  and  third  members  of  (203)  may 
also  be  obtained  by  the  model  form  of  eq.  (176).  For  instance, 
the  projection  of  PQ  on  q.,qs  directed  by  qiPi  is 

= ■ =  g.//3  •  Fa,. 

Again  let  P  be  as  in  (203)  and  L  =  q^q., ;  then 

=  Pi  -P-iPiL  +2h -PaPiL  +P3-2hP2L  ) 

the  results  being  found  as  in  previous  cases. 
Multiply  the  first  of  (204)  into  Q ;  then 


LPQ 


qyP  qxQ\ 

qoP  q^M 


(205) 


Let  Q  and  R  be  two  planes  intersecting  in  L,  and  substitute 
QR  for  L  in  (204)  ;  then,  by  (205), 

PQR  =pi  ■PipiQR  +i>2  -PiPiQR  +Ps  -PiP^QR  ^ 
Pi  PiQ  PiR 


P2  IhQ  IhR 

Ps  PiQ  PsR 


(206) 


If  iS  be  a  fourth  plane,  multiply  (206)  into  it,  and  we  have 


PQRS  = 


PiQ  PiR  2hS 
PiQ  P2R  P2S 
PsQ  PsR  PsS 


(207) 


Of  course  three  other  equal  expressions  could  be  Avritten  in 
terms  of  points  in  the  other  planes. 

In  (205)  let  P=  \2h  and  Q  =  \p2,  and  we  have 


qiq2\PiP2=PiP2\qiq2  = 


Pi\qi  pm 
pMi  p-2\q2 

as  in  eq.  (117).     In  (206)  let  Q  =  \qi  and  R  =  \q2;  then 

P\P2Pz\q\q2 = 2h'2hP3\qiq2+P2'PzP\\qiq2+Pz  -Pip^m^ 
Pi  Pi\qi  2h\q2 


P2  P2\qi  P2\q2 
Ps  Pz\qi  Pz\q2 


(208) 


(209) 


Chap.  II.]  MULTIPLICATION.  71 

Finally  in  (207)  give  Q  and  R  the  above  values,  and  let 
S  =  1^-3 ;  then 

Pi\Qi  Pi\<l2  Pl\q^^ 
PiPiPslQiQ-fls  =  P2\qi  .P-hi  P^\% (210) 

Pi\qi  p&\(i2  Pz\q& 

By  eqs.  (201)  and  (210)  it  may  be  shown  that  we  have 

jPlP2P3P4-giM394  =  [i>l|9oi\!5'*P3!<?3,i>4|g4],        •         •        ■         (211) 

in  which  the  second  member  is  a  determinant  formed  on  the 
plan  of  (210),  of  which  the  quantities  given  make  up  the  first 
diagonal. 

In  all  these  equations  points  may  be  put  for  planes  and 
planes  for  points  without  affecting  their  validity.  Also,  be- 
cause of  the  homogeneity  of  the  equations  in  all  the  points 
involved,  these  points  may  have  any  weights  we  please. 

67.  Normal  form  of  the  screw.  Returning  now  to  the  sub- 
ject of  Art.  61,  we  propose  to  show  that  by  properly  choosing 
the  position  of  the  line  part  of  S,  the  screw  can  be  reduced  to 
a  line  and  a  perjjendicidar  plane-vector.  The  complement  as 
used  in  treating  screws  will  refer  to  a  unit  normal  vector  system, 
so  that  |c  will  be  a  plane-vector  ±  to  c  and  having  the  same 
tensor.     We  have,  from  Art.  61, 

S  =  eoSc  +  %{p-  eo)£  =  q%t  -  {q  -  eo)2c  -|-  2(i>  -  eo)c.  (212) 

Write,  for  convenience, 

2c  =  a,  q  —  eo  =  p,  and  '${p  —  eo)£=\/3 ; 

.-.  S  =  qa-pa  +  \(3 (213) 

The  condition  that  the  plane- vector  \p  —  pa  shall  be  perpen- 
dicular to  a  is 

(1^  —  pa)  |a  =  0  =  |)8a  —  pa|a  =  |a  •  pa  —  |ay3, 

whence        hlP^J^ (214) 

a-  a- 

Comparing  the  first  member  of  (214)  with  eq.  (176),  it 
appears  that  it  is  the  orthogonal  projection  of  p,  or  q  —  Co,  on 


72  DIKECTIONAL   CALCULUS.  [Art.  68. 

a  plane  J.  to  a.  Hence  the  second  member  gives  the  length 
and  direction  of  this  projection  in  terms  of  known  quantities ; 
that  is,  it  is  the  vector  perpendicular  between  the  lines  eoSe 
and  q%€.     We  have  by  (189)  and  (214) 

a(|a  •  pa)       o-{,o-P  ■  i^)        «(  —  o-  •  p|a  +  p  •  a-) 


.2 


ajayS       ]a  •  a]/? 
a-  a- 

Substituting  this  value  of  ap  in  (213),  it  becomes 

>S^ga  +  ^.la^.y2c  +  ^l^ff^^.|2c,    .     .     .     (215) 
a^  (20- 

and  the  required  reduction  is  accomplished. 

68.   Product  of  two  screws.     Let  the  screws  be 

Si  =  efy  +  ciirji  =  CjCi  -\-  ai€i  and  S2  =  CjCo  -f-  a^i^o  =  62^2"!"  '^2k2> 

in  which  Oj  and  as  9-^6  scalars,  called  by  Ball  the  j)itches  of  the 
respective  screws.    Then 

SiS2=  (^iCi  +  ttiiyi)  (e2^2  +  f'2'72)  =  6l^l&2^2-^(^^l^lV2'^^l'^2^Vl'^'^l'^2VlV2' 

Now  this  is  a  progressive  product,  each  term  being  the  prod- 
uct of  two  lines,  and  scalar;  the  two  lines  in  the  last  term 
being  in  the  plane  at  00,  they  intersect,  and  their  product  is 
therefore  zero.     (See  also  Art.  23.)     Further,  by  eq.  (45), 

ejCiT^a  =  C1T72  =  cilca  =  «2|ci  =  «2'7i  =  e2€2i7i ; 
hence  the  product  becomes 

/S'i/S2  =  6lCl62«2  +  (ai  +  a2)ei|£2 •       •        (216) 

If  >S'2  =  Si,  we  have 

Si'  =  2ai€il     .....' (217) 

If  Si  reduce  to  a  Ime,  ttj  must  be  zero,  as  appears  from  the 
value  at  the  beginning  of  the  article ;  hence 

S'=0 ; (218) 

is  the  condition  that  a  screw  shall  reduce  to  a  line. 


Chap.  IT.]  MULTIPLICATION.  73 

69.  The  product  pS  =pee  +  ap-q  is  evidently  a  plane  through 
the  common  line  of  the  planes  pec  and  p-q.  We  wish  to  show 
that  we  have 

Sjh  •  Sjh  =  iSpiPi  •  S  —  ajhPi  •  ^ (219) 

Take  two  lines  pp'  and  qq'  whose  sum  is  S ;  then 

Spi  •  Sp2  =  ipp'  +  qq')Pi '  {pp'  +  qq')p2 

=PP'Pi  -pp'Pi  +pp'Pi  •  qq'P2  +  qq'pi  -pp'Ps  +  qQ'Pi '  qg'Pi 

=pp'PiP2 '  pp'  +  qq'piPi  •  qq'  +pp'  -pm'Pi  +p'pi  -pqq'Ps 

-\-PiP  •p'qq'p2+qq'  •Pipp'p2+q'pi  •  qpp'P2+Piq  •  q'pp'P2 

=  pp'piP2 '  S  +  qq'PiPi  •  s 

+2h(p  •p'qq'P2  —p'  •pqq'P2  +  q  •  q'pp'ih  —  q'  •  qi^p'ih) 

=  SpiP2  •  S  -P1P2  -pp'qq' 
=  SpiPs  •  S  —P1P2  •  «e«Ic 
=  Sp^p2  •  S  —  apiP2  ■  e-. 

In  the  above  we  have  used  eqs.  (197)  and  (204),  and  also 
the  fact  that  pp'qq'  is  constant  whatever  the  lines  may  be  so 
long  as  pp'  +  qq'  =  S,  for  S^  =  2 pp'qq'  =  2a£-,  by  (217).     " 

We  easily  find  in  the  same  way, 

S'pS  =  ae-p, (220) 

Spx '  Sp2  •  Sps  =  at^{2h  •P2PzS  -ifih -PsPiS  -hPs •PiP2S).   (221) 
In  all  these  equations  planes  may  be  substituted  for  points. 

70.  We  give  here  the  Quaternion  equivalents  of  some  of 
our  vector  expressions ;  of  course  there  are  no  such  equivalents 
for  point  expressions. 

Cile2  =  —  S€i€2,  |cie2  =  FciCg, 

ti«2e3  =  —  'S'eie2%  ^2^^h  =  —  Ve^  FcaCg, 

CiCzIesC*  =  —  S'  Ci£2Fc3e4,  CjCa  •  c^e^  =  V-  FeiC2F£3C4, 

C1C2  •  ^U  •  ^5^  =  —  S  •  Fcie2  Fe3e4  FejCg. 
The  superior  simplicity  of  Grassmann's  notation  is  evident  at 

a  glance,  and  the  interpretation  of  the  expressions  is  as  much 

simplified  as  their  form. 


74  DIRECTIONAL   CALCULUS.  [Art.   71. 

71.  ExERCiSE.s. —  (1)  If  r]i  and  t;,  ^^"6  two  plane-vectors, 
and  Pi  and  P2  ^-re  two  point-plane-vectors,  show  that  the  bisec- 
tors of  the  diedral  angles  between  them  are 

rjiTr],  ±  r].Tr}i  and  P^TP,  ±  PoTPi 

respectively.  If  r?i  =  e^co,  rj.,  =  £,e.,,  Pj  =  jhlhlh,  and  P^  =2\2hlh'> 
these  become 

ci(f2^«iC3  ±  csT'tic-')  and  Pi^PiiihTj^oPilh  ±  P:'T]\>PiP-2)  ; 

or,  if  we  write  A^,,  A^,  A^  Ag  for  the  double 
areas  of  the  faces  of  the  tetraedron  oppo- 
site j>0)  Pif  6tc.,  the  expressions  become 

£1(^2^2  ±  ^£3)  and  PoPi(A2P2  ±  A^p^). 

(2)  Show  that  Pq(AiPi  ±  A2P2  ±  A^Ps) 

and  AiCi  ±  A^f.^  ±  A^^  are  trisectors  of  the  triedral  angle  at  p^ ; 
that  is,  that  the  first  expression  is  the  common  line  of  the 
bisecting  planes  through  p^^Px,  Pf^p^  and  P(sPzi  while  the  second 
is  II  to  it. 

(3)  The  trisector  of  a  triedral  angle  of  a  tetraedron  pierces 
the  opposite  face  in  a  point  such  that,  if  it  be  joined  by  right 
lines  to  the  vertices  of  the  tetraedron  that  are  in  this  face,  the 
triangles  thus  formed  are  proportional  to  the  adjacent  faces. 

(4)  The  bisecting  plane  through  one  edge  of  a  tetraedron 
divides  the  opposite  edge  into  segments  which  are  proportional 
to  the  adjacent  faces. 

(5)  The  twelve  bisecting  planes  of  the  diedral  angles  of  a 
tetraedron  pass  six  by  six  through  eight  points  which  are  the 
centers  of  the  inscribed  and  escribed  spheres. 

(6)  The  twelve  points  in  which  the  edges  of  a  a  tetraedron 
are  cut  by  the  bisecting  planes  of  the  opposite  diedral  angles 
fix  eight  planes,  each  of  which  passes  through  six  of  them. 

(7)  Using  Af„  Ai,  etc.,  as  in  exercise  (1),  show  that 

A'  =  a;'  +  A./  +  A'  -  2 ^2^3 cos  <^^-2AoAi  cos  <^' 
—  2  ^1^2  cos  <^-. 


Chap.  TI.]  MULTIPLICATION.  75 

(S)  Show  that  if  e^,  gj,  62,  e.  are  non-coplanar  points,  and 
€0,  Ci,  eJ,  eJ  divide  the  lines  e^ei,  6163,  6263,  and  e^eo,  so  that 

then  will  e,/,  e/,  p^'j  ^s'  be  co-planar. 

(9)  If  a  plane  cut  the  faces  of  a  tetraedron  e^e-fi^^..  in  the 
lines  Z/Q)  -^15  ^2»  -^s*  -Z>o  lyij^g  i^^  the  face  opposite  to  eo?  etc., 
then  Ave  shall  have  the  relations 

^^  ^O-'-'.'J  *  ^l-'^O  *  ^2-*-'l  *  ^3-"2' 

(10)  By  interchanging  planes  and  points  derive  the  recipro- 
cal propositions  to  (8)  and  (9). 

(11)  If  two  tetraedra  e^e^eoe^  and  ef^e^e^^-l  are  so  related  that 
the  right  lines  through  corresponding  vertices  all  meet  in  one 
point,  then  will  the  corresponding  faces  cut  each  other  in  four 
coplanar  right  lines. 

(12)  If  U],  Wgj  etc.,  are  scalars,  and  L^,  L.,,  etc.,  lines,  and  we 
have  the  relation 

n^Lx  +  '^2-^2  +  W3Z/3  +  iiiLi  =  0, 

then  any  straight  line  that  cuts  .three  of  these  lines  will  also 
cut  the  fourth,  and,  consequently,  Xj,  L2,  Xg,  A  are  generators 
of  the  same  system  of  a  skew  quadric. 

(13)  The  perpendiculars  from  the  vertices  of  a  tetraedron 
on  the  opposite  faces  are  generators  of  the  same  system  of  a 
skew  quadric. 

(14)  The  six  planes  through  the  middle  points  of  the  edges 
of  a  tetraedron  X  to  the  respective  edges  meet  in  one  point. 
If  €x,  €2?  ^3  ^-re  the  vector  edges  of  the  tetraedron  drawn  out- 
ward from  €(,,  and  p  is  the  vector  from  €q  to  the  common  point 
of  the  planes,  then 

p  = (|e2C3  •  «i-  +  ksCi  •  ^2-  +  leiC2  •  ^3"). 


76 


DIRECTIONAL   CALCULUS. 


[Art.  71. 


(15)  The  lines  joining  the  corresponding  vertices  of  a  tetrae- 
dron  and  its  complementary  tetraedron  are  generators  of  the 
same  system  of  a  skew  quadric.  State  the  reciprocal  propo- 
sition. 

(16)  The  center  of  gravity  of  the  faces  of  a  tetraedron 
coincides  with  the  center  of  the  sphere  inscribed  within  the 
tetraedron  formed  by  joining  by  right  lines  the  mean  points  of 
the  faces  of  the  first  tetraedron. 

(17)  There  are  given  six  lines  L^,  Xg,  X,,,  e^e^,  e^e-z,  e^.^ ; 
planes  pass  through  L^,  L~i,  L^,  and  cut  e-fii,  e.^.2,  e^e^  respec- 
tively, in  points  which  move  along  these  lines  uniformly  at 
rates  v^,  v^,  v^ ;  find  the  locus  of  the  common  point  of  these 
planes. 

;     (18)  If  Pi  =  S^je,  P2  =  Somie,  P,  =  ^n\e,  show  that 


1 

mo 


1 

k 

mi 
n. 


1 

k 

mo 
n., 


=  0 


is  the  condition  that  they  shall  have  a  common  point  at  oc ; 
that  is,  be  all  ||  to  one  line. 

(19)   Show  that  the  condition  that  Pj  and  P2  of  the  last 
exercise  shall  be  parallel,  or  have  a  common  line  at  cc,  is 


1 

1 

1 

1 

h 

h 

h 

= 

^1 

mo 

Wi 

m^ 

m 

1 

1 

I, 

^3 

= 

wig 

mg 

1 

1 

1 

1 

1 

1 

k 

^3 

^0 

= 

^3 

h 

^1 

=  0. 

m^ 

m^ 

mo 

Wis 

mo 

mi 

(20)  Show  that,  if  Pj  and  P^  be  parallel,  then  [PjPo  is  a  line 
through  the  mean  point  of  the  reference  tetraedron. 

(21)  If  any  plane  be  drawn  through  the  middle  points  of 
two  opposite  edges  of  a  tetraedron,  it  will  divide  the  volume 
of  the  tetraedron  into  two  equal  parts. 

(22)  Show  that  piLiLiL-^p^^  —  p^L^L^L^pi,  and 

L^PiLrPX.,  =  -  L.PJj.PJji. 


Chap.  II.]  MULTIPLICATION.  77 

(23)  Prove,  by  eq.  (159),  the  relation 

ih'  +  ^2'  +  h^  (wf  +  WI2'  +  mg-)  >  (Zim,  +  hm,  +  hm^y. 

(24)  Prove,  by  eq.  (193),  the  formula  of  spherical  trigonome- 
try cos  a  =  cos  b  cos  c  +  sin  6  sin  c  cos  A. 

(25)  Show  that  I,  |tc,  and  tc«[i  are  three  mutually  perpen- 
dicular vectors,  no  matter  what  the  directions  of  i  and  e  may  be. 

(26)  Show,  by  eq.  (211),  how  to  express  the  product  of  two 
determinants  of  the  fourth  order,  as  a  determinant  of  the  same 
order. 

(27)  Show  that  SP  =  p,  •  P2P3S  -\-  p,  •  PsP\S  +  2h '  P1P2S,  if 
P  =  PiP2P3 ;  and,  hence,  that  Sj)i  •  Sp2  •  Sps  =  at-  •  S  -PiPiPz. 


Chap.  III.]       APPLICATIONS   TO   PLANE  GEOMETRY.  87 


CHAPTER   III. 

APPLICATIONS   TO   PLANE   GEOMETRY. 

72.  In  the  present  chapter,  since  plane  space  is  under  con- 
sideration, we  shall  have  constantly  : 

The  product  of  two  vectors  a  scalar  qiiantity; 
The  product  of  three  points  a  scalar  quantity. 

Also,  if  Co,  «!,  €3  are  reference  points,  and 

£j  =  ei  — e„,   co  =  e.2  —  eo, 
we  shall  have  the  relation 

606162  =  60(61  —  60)  (60  —  60)  =  eoCiCo  =  £1^2  =  ^1^2  +  ^260  +  et,ei  =  1 ; 
and  furthermore,  if  p  be  any  unit  point  at  a  finite  distance, 

p\  (eo  +  61  +  62)  =p(eieo  +  6360  +  ^061)  =2^^1(2  =  ci«2  =  L      (222) 

In  this  equation  of  course  peit^  is  a  combinatory  product  of 
the  point  and  two  vectors,  and  therefore  not  the  same  as  ]) 
times  the  scalar  cjCg.  That  p\  (60  +  61  +  Co)  =  1  appears  also 
from  eq.  (105),  viz. : 

p  =  eo-  2y\eo  +  61  •  pCi  +  62 .  pjea, 

which  requires  the  sum  of  the  coefficients  of  the  e's  to  be  unity. 
We  shall  have  frequent  occasion  to  use  the  mean  point  of 
the  reference  triangle,  and  shall  designate  it  by  e,  so  that 

3e  =  e,  +  e,  +  e.^ (223) 

and  eq.  (222)  becomes  

3i)le  =  l (224) 


88  DIRECTIONAL   CALCULUS.  [Art.  73. 

The  equations  of  curves  in  plane  space  may  appear  under 
any  one  of  the  six  following  forms  :  — 

r  Expressed  in  points. 
Non-scalar  equations.  <  Expressed  in  lines. 

(  Expressed  in  vectors. 

r  Expressed  in  points. 
Scalar  equations.  ■<  Expressed  in  lines. 

(  Expressed  in  vectors. 

73.   The  non-scalar  equation 

p  =  ze^  +  xci  +  2/62  =  eo  +  X (ei  —  Co)  +  2/  (^2  -  ^o) ,       (225) 

the  third  member  being  obtained  by  the  elimination  of  z,  by 
the  aid  of  the  relation  x-\-y  +  z  =  l,  which  always  exists 
because  we  use  only  unit  points,  may  be  called  the  equation  of 
our  plane  space;  for  by  giving  suitable  values  to  the  scalar 
variables  p  may  be  moved  to  any  point  of  this  space.  The 
corresponding  scalar  equation  is 

606162^  =  0, (226) 

which  is  simply  the  condition  that  p  shall  lie  in  the  plane  eo^i^s- 
If  a  single  condition  be  given  between  the  scalar  variables 

in  eq.  (225),  such  as  f{x,  y,  z)  =  0,  or  f{x,  y)  =  0,  then  2>  will 

vary  according  to  a  fixed  law,  and  will  therefore   move   on 

some  curve. 

Let  L=\p;  then 

L  =  z\eo  +  x\ei  +  y\e2 (227) 

may  be  any  line  in  the  plane  60^162 ;  but  if  a  relation  exist,  as 
above,  between  z,  x,  and  y,  then  L  will  move  according  to  some 
fixed  law,  and  will  envelope  a  curve. 

Writing  in  (225),  p  —  e^  =  />,  Cj  —  ?„  =  cj,  Pj  —  ^o  =  %  we  have 

p  =  x,,  +  ye,, (228) 

a  vector  equation  which  will  represent  a  curve  when  a  relation 
exists  between  x  and  y,  p  being  regarded  as  always  drawn  out- 
wards from  a  fixed  origin. 


Chap.  Ill]       APPLICATIONS   TO   PLANE   GEOMETRY.  89 

74.   The  equations 

J)  =  zcty  +  xei  +  wfio ) 

,  n    h (229) 

ix  +  my  +  nz  =  0   } 

taken  together,  represent  a  right  line ;  for,  eliminating  z,  we 
have 

p  =  -[x (ngj  —  ?eo)  +  2/ ("^2  —  me,^)  ]  ;      •     (230) 

so  that  2^  lies  on  the  right  line  through  the  two  points  nci  —  le^ 
and  7162  —  tnef).  Multiplying  by  (nei  —  leo)  (ne^  —  mefj),  we  obtain 
the  corresponding  scalar  equation 

(«ei  —  le^)  (ne2  —  meo)})  =  0,  -v 
or  ^(716162  +  ^6260  +  ^16061)  =0,  >-....     (231) 

or  p\  (n6o  +  ?6i  +  me.,)  =  0.  ^ 

If  l  =  m  =  n,  this  equation  becomes  that  of  the  line  at  oo, 
viz.: 

p\e  =  0', (232) 

for  the  points  wei  —  le^  and  ??62  —  me,,,  in  which  the  line  cuts 
the  reference  lines  6061  and  6062,  are  in  this  case  at  00 . 

The  equation  of  a  line  through  any  two  points,  p^  and  p2, 
may  be  written 

p  =  xpi  +  (1  —  x)p2=p2  +  a;(pi  —po),        (233) 

and  the  corresponding  scalar  equation,  found  by  multiplying 

by  IhP-z,  is 

PPiP2  =  0 (234) 

In  general,  the  equation 

pL  =  0 (235) 

is  the  scalsiT  point  equation  of  a  line,  if  L  be  constant  and  p 
vary,  and  the  scalar  line  equation  of  a  point,  if  j)  be  constant 
and  L  vary.  Thus,  the  complementary  equations  to  (231). 
(232),  and  (234)  are 


90  DIRECTIONAL   CALCULUS.  [Art.  75. 

i(«eo  4- ?^i  +  we,)  =  0, (236) 

ie  =  0, (237) 

LL,L2=0, (238) 

which  are  line  equations  of  the  points  ne^  +  Zej  +  meo?  ^,  and 
L1L2,  respectively. 

If  we  have  such  an  equation  as  jipip,,  =  C,  C  being  a  scalar 
constant,  it  may  always  be  rendered  homogeneous  in  p ;  for, 
by  eq.  (224),  3p\e  =  1,  so  that  we  may  write 

PP1P2  =  3  Cp\e, 

or  p(PiP2-SC-\e)  =  0, (239) 

which  is  a  line  through  the  common  point  of  pip^  and  the  line 
at  00,  and  is  therefore  parallel  to  2hP2- 

For  vector  equations  of  right  lines  we  have 

'^  [, (240) 

and  (p  — £i)«2  =  0) 

the   scalar   and  non-scalar  forms  of  the  equation  of  a  line 
through  the  end  of  cj  parallel  to  c,. 

Also  p  =  €i-hx{c2-e,),  I 

and  (/J  —  €1)  (c2  —  fi)  =  0,  OT  p  (e.^  —  £1)  =  ci£2,  ) 

for  the  two  forms  of  the  vector  equation  of  a  line  through  the 
ends  of  ci  and  cg  drawn  outwards  from  the  origin. 
The  equation 

pe  =  C (242) 

is  that  of  some  line  parallel  to  c,  while 

p\e  =  C.     . (243) 

is  that  of  some  line  perpendicular  to  c,  as  is  easily  seen  from 
the  meanings  assigned  to  pe  and  p\e  in  Chap.  II. 

75.  Transformation  of  scalar  equations  from  a  point  system 
to  a  vector  system,  and  vice  versd.  Take  e^  for  the  origin  of 
vectors,  and  write  p  —  eQ  =  p.,  2h  —  %  =  ^\}  ^tc,  the  difference 


Chap.  HI.]       APPLICATIONS   TO   PLANE   GEOMETRY.  91 

between  each,  fixed  point  and  the  origin  being  equal  to  some 
constant  vector.  Thus  to  transform  (234)  to  a  vector  system, 
we  have 

PPllh  =  (^0  +  p)  (^0  +  Cl)  (^0  +  f-l)  =  eo(pei  +  Clfo  +  Cop)  =  0. 

The  term  pciCj  vanishes  by  Art.  21,  being  the  planimetric 
product  of  three  vectors ;  also  eop^i  =  p^i>  ^tc,  hence  the  equa- 
tion becomes  /3(c2  — ci)  =  ciCg?  the  same  as  (241).  Since  in 
changing  from  a  point  to  a  vector  system  we  have  dropped 
the  point  Cq  from  each  term,  it  follows  that  in  the  reverse 
change  we  must  first  multiply  each  term  by  some  fixed  point. 
Thus  to  change  (p  —  cj)  C2  =  0  to  a  point  equation  we  have 

(p  -  ei)c2  =  eo(p  -2h)  (P-2  -  eo)  =  eo(i^  -Pi)Po  =  0. 

76.  ExEKCiSES.  —  (1)  Find  the  equations  of  right  lines 
satisfying  the  following  conditions  :  — 

Passing  through  pi  and  parallel  to  Li ; 
Passing  through  j9i  and  parallel  to  e ; 

Passing  through  the  common  point  of  two  right  lines  and 
having  a  given  direction ; 

Passing  through  the  common  point  of  two  right  lines,  and 
also  through  the  common  point  of  two  other  right  lines ; 
Passing  through  the  end  of  cj  perpendicular  to  e^. 

Am.  {p  —pi)Li  =  0,  p2?i€  =  0,  peLiL.2  =  0, 
p{L,L,){L,L,)  =  0,  (p-£i)Ic2  =  0. 

(2)  Interpret  the  equations  obtained  by  putting  lines  for 
points,  and  points  for  lines,  in  the  first  four  results  of  the  pre- 
vious exercise. 

(3)  Find  the  common  points  of  the  following  pairs  of  right 
lines, 

Sp\P,  =  0\        ipprp,  =  0)         Uci=Ci)        ^p\e^=C,l 

lp\P2=or   \pqxq-2=^y    lpe,=  ar  lp\e,=  aj' 

(4)  Find  the  condition  that  the  three  lines  i>|i>i=0,  p\p2=0, 
p\Ps=0  shall  have  a  common  point;  also  the  lines  p[ei=  Ci, 
p|c2  =  Cj,  p\e.i  =  Cg. 

Ans.  xtiPiPz  =  ^>  and  CiCjCs  +  C<a^i  +  Cs/^x'^i  =  0. 


92  DIRECTIONAL  CALCULUS.  [Art.  77. 

(5)  Show  that  the  common  point  of  the  two  lines  ppiP2=  C^ 

PP2Ps=Ci  is   P2+-—-lCs(p^i-p-2)  +  C,(2h-2h)l 
PuhPs 

(6)  Show  that,  if  PiP^Pz  =  Ci  +  Cj  +  63,  then  the  three 
lines  PP1P2  =  C3,  pPiPz  =  Gi,  PlhPi  =  ^2  have  a  common  point. 

(7)  Show  that  if  the  equations  of  three  lines,  on  being  mul- 
tiplied by  any  constants  and  added,  vanish  identically,  that  is, 
for  all  values  of  p  or  p,  then  the  lines  have  a  common  point. 
Show  also  that  the  results  in  exercises  (4)  and  (6)  are  in 
accordance  with  this. 

(8)  Find  the  condition  that  the  three  points  whose  line 
equations  are  L\Li  =  0,  Xjij  =  0,  LL^  =  0  shall  be  coUinear. 

(9)  Show  that  the  perpendiculars  from  the  point  e  on  the 
lines  whose  equations  are  pLi  =  0,  j^L^  =  C,  {p  —2h)Pi2h  =  % 
are  respectively  of  the  length  -^,    iklZ^    2hP-2{e-2h), 

(10)  Find  the  vector  perpendiculars  from  the  origin  on  the 
lines  ep=C  and  c|p  =  C.  Also  from  the  end  of  e'  on  the  same 
lines.  ^^    O  ,_    Ce     O-ce'   ,      C-ck' 


^.        ^'  ^.'  £?  "  £? 

(11)  If  Li=piei,  L2=P2€2,  etc.,  show  that  T1,L  =  T%i. 
We  have,  by  Art.  61,  since  the  lines  are  all  in  one  plane, 
"XL  =(^0  +  a)  Se ;  but  60  +  a  is  a  unit  point,  hence  TSZ  =  T2c. 

(12)  Show  that  T{L-\-^C\e)=  TL. 

77.  If  ii  and  L2  are  two  straight  lines,  then  the  equation 
LxP  •  L2P  =  0  represents  the  two  lines  simultaneously,  for  it 
is  satisfied  whenever  p  lies  on  either  of  the  lines.    The  equation 

L^p-L.,p  =  C (244) 

represents  a  locus  that  evidently  differs  less  from  being  the 
two  lines  L^  and  L^,  the  smaller  C  is ;  also,  when  p  is  indefi- 
nitely far  from  L^,  it  is  indefinitely  near  to  L2,  and  vice  versd. 
The  locus  is  of  the  second  order ;  i.e.  it  is  cut  in  tAvo  points  by 


Chap.  HT.]       APPLICATIONS  TO  PLANE  GEOMETRY.  93 

a  right  line ;  for  let  p  =  e-\-X€  be  the  equation  of  some  line j 
then,  substituting  this  value  oij)  in  (244),  we  have  a  quadratic 
in  X  determining  two  points  in  which  the  line  cuts  the  curve. 
The  locus  raust  therefore  be  a  hyperbola.  If  C  be  positive, 
LiP  and  Lop  must  have  like  signs ;  hence  p  must  be  on  the  same 
side  of  ij  and  of  L^,  i.e.  in  the  exterior  angle,  while,  if  C  be 
negative,  x>  must  be  in  the  interior  angle.  Thus  for  the  same 
numerical  value  opposite  signs  of  C  correspond  to  a  primary 
and  conjugate  hyperbola. 

The  complementary  equation 

p,L-pX  =  C (245) 

represents  the  reciprocal  curve  to  (244).  ^Vhen  C=0,  it 
represents  the  two  points  p^  and  p2  and  their  connecting  line ; 
for  it  is  satisfied  when  L  passes  through  p^  or  p.2,  or  through 
both.  "When  C  is  positive,  p^L  and  p^L  must  have  like  signs, 
and  hence  L  must  not  pass  between  pi  and  p.,',  if  C  be  negative, 
L  must  always  pass  between  pi  and  p^r  The  curve  enveloped 
by  L  is  of  the  second  class,  i.e.  two  tangents  can  be  drawn 
from  any  point ;  it  is  therefore  a  conic. 

78.  It  is  easily  seen,  as  in  the  last  article,  that  the  order  of 
the  curve  represented  by  any  scalar  equation  in  terms  of  p, 
i.e.  the  number  of  points  in  which  it  can  be  cut  by  a  right  line, 
is  simply  the  degree  in  p  of  the  term  of  highest  degree  in  the 
equation. 

The  equation 

AL.iP  .  Lsp  +  BL^p  •  Ai>  +  CLip  .  L.p  =  0,     .     (246) 

in  which  A,  B,  C  are  scalar  coefficients,  represents  a  curve  of 
the  second  order  passing  through  the  points  LiL.>,  L-iL^,  L3L1 ; 
for  each  term  is  of  the  second  degree  in  p,  and  the  equation  is 
satisfied  when  2>  is  on  any  two  of  the  lines  simultaneously. 
The  complementary  equation 

Apjj '  psL  +  BpsL  •  piL  +  CpiL '  P2L  =  0   .     .     (247) 

causes  L  to  envelop  a  curve  of  the  second  class  tangent  to  the 
three  lines  j^ip^  P2P3)  PaPi  >  ^^^  i*  is  satisfied  when  L  passes 


94  DIRECTIONAL  CALCULUS.  [Art.  79. 

through  any  two  of  these  points  simultaneously.  As  an  exer- 
cise let  the  student  interpret  the  following  equations,  k  being 
a  scalar  constant : 

PiPiP  ■  PslhP    =  khPiP  '  PzPiP, 

P1P2P  -PsPiP   ^  PlPiP  -PiPsP 

P1P2P0  ■  PsPiPs  ~ PlPiP5  ■  P2PsP5 

JjiIjoIj  '  Ju^ljilj    =  kljiL/iju  •  Ij^L^L^ 

79.  Differentiation.  Before  proceeding  to  the  general  treat- 
ment of  equations  of  the  second  degree  in  p  and  7),  we  will 
consider  the  question  of  differentiation  as  applied  in  this  cal- 
culus. 

Let 

p  =  zeo  +  xei  +  ye2  =  e„  +  x(ei  —  go)  +  2/  (^a  —  ^0)  =  ^'o  +  xe^  +  yt^, 

which  implies  that  x  +  y  -{-z  =  l,  as  we  always  assume. 

If  p  move  from  point  to  point,  it  is  a  function  of  the  time, 
as  are  also  x  and  y ;  hence 

#  =  e,^-fc,^ (248) 

dt         dt         dt  ^       ^ 

Thus  the  differential  coefficient  of  a  point  is  a  vector.  Also, 
since  p=p  —  eQ, 

^  =  ^ (249) 

dt      dt  ^       ^ 

If  Tcj  =  Tca  =  1,  tile2  =  0>  and  a  relation  subsist  between  x 
and  y  such  as  f{x,  y)  =  0,  so  that  2>  moves  along  some  curve, 

<S)=I •  •  (^^) 

whence  T^  =  l, (251) 

ds       '  ^      ^ 


CiiAP.  TIT.]      APPLICATIONS   TO   PLANE   GEOMETRY. 


95 


Let  L=\p=\e(i-\-  xe.j.{eQ  +  e,)  —  yei{e.,  +  e^,  so  that  L  is  sub- 
jected to  the  same  condition,  z  -\-x-\-  y  =  \,  that  p  is,  which, 
however,  affects  only  its  length,  and  not  in  any  way  its  position ; 


then 


—  =  62(^0  +  eO  -  ^(?i(e2  +  eu)- 


(252) 


clL 


Multiply  —  by  e,  and  we  have 
dx 

e-—  =  e^e.ei  +  e,e./o  —  --{e^e^e.^  +  e.>e^e^,)  =  0  ;  (253) 
ax  ax 

hence  —  is  a  line  through  the  mean  point  of  the  reference 
do; 

triangle. 

By  the  figure  it  is  evident  that 

p—p'  or  p  —  p'  is  a  chord  of  the 

curve  which  is  the  locus  of  2> ;  as 

p'  approaches  p,  p  —p'  approaches 

the  tangent  at  p  in  direction,  and 

at  the   limit   has   this   direction ;   ^0 

hence 

limit  of      P-P'      =^^l 
T(p  —p')      Tdp     ds 


(254) 


is  a  unit  vector  along  the  tangent  at  j)- 


dL 


Similarly,  if  L  envelops  some  curve,  • is  the  limit  of 

— — — -  as  L'  approaches  L.     But  L  —  L'  is  always  a  line 

through  the  common   point   of   L  and   L',  which   ultimately 
becomes  the  point  of  contact  of  L  with   the   curve.     Hence 

is  a  unit  line  through  the  point  of  contact  of  L  and  the 

mean  point  e. 

If  a  scalar  equation  in  ^9,  L,  or  p  be  differentiated,  it  will 
necessarily  become  a  homogeneous,  linear  function  of  dp,  dL, 
or  dp,  and  thus  independent  of  the  length  of  dp,  dL,  or  dp ;  we 


96  DIRECTIONAL  CALCULUS.  [Art.  80. 

may  therefore,  if  we  please,  regard  these  not  as  infinitesimals, 
but  as  finite  in  length.     Take  for  instance  the  equation 

Ph'Ph  -\-pLi  -\-pL2  -pLs  -pLi  =  0 ; 
differentiating,  we  have 

dp\eQ  •  p|ei  +  p\eo  •  dp\ei  +  dpL^  +  dpLi-pL-i-pLi 

+  pL.2  •  dpLy,  •  pL^  +  pL^  •  pL:^  •  dpLi  =  0 ; 

and,  as  dp  appears  once,  and  only  once,  in  each  term,  it  is  evi- 
dent that  its  length  may  be  taken  as  great  or  as  small  as  we 
please  without  affecting  in  any  way  the  meaning  of  the  equa- 
tion. 

80.  Examples  of  differentiation.  As  shown  by  the  example 
just  given,  the  process  of  differentiation  does  not  differ  in 
principle  from  that  of  ordinary  algebraic  equations ;  we  have 
only  to  pay  attention  to  the  alternative  law  of  multiplication. 

dipPiLiPiP)  =  dppiLiPop  +  pjhLiPs^P    \ 

=  dppiLiPiP  +  2hdp  •  PlhLi  > .      .     (255) 
=  dp(piLiP2P+2hLiPPi)    ^ 
d{peq)=dpeq -{-pedq  =  e{qdp—pdq).   .     .     .     (256) 
d(f^)  =  d(p\p)=:dp\p  +  p\dp  =  2p\dp..     '.    .     .     (257) 

d{e\py  =  n{e\pY~'e\dp (258) 

(ZrV  =  2TpdTp  =  d(f^)  =  2p\dp. 

.-.  dTp  =  P-^  =  Up\dp (259) 

dp  =  d{TpUp)  =  UpdTp-{-TpdUp  I 

=  Up'Up\dp+TpdUp  I    •    '    '    •     ^      ) 

Also,  by  eqs.  (189)  and  (260), 

Updp'\Up  =  dp-Up'dp\Up  =  TpdUp.  .    .    .     (261) 

The  student  will  find  it  interesting  to  examine  the  geomet- 
rical significance  of  the  last  three  equations. 


Chap.  III.]       APPLICATIONS  TO  PLANE  GEOMETRY.  97 

81.  Tangent  and  normal.    If  the  equation  of  a  curve  be 
given  in  the  form 

p  =  <f>{x)  =  X€,+f{x)-e.2, (262) 

then,  as  -^  is  a  vector  along  the  tangent  at  the  end  of  p,  if  we 
dx 

let  o-  be  a  vector  to  any  point  of  the  tangent,  we  have  for  the 
equation  of  the  tangent  at  the  end  of  p,  u  being  a  variable 

a  =  p  +  u^  =  ^{x)  +  u<f>\x) (263) 

dx 

Multiplying  by  -^,  we  have  the  scalar  form 
dx 

((r-p)dp  =  0  =  {<r-<t>{x))<l>'(x).  .     .     .     (264) 

If  V  be  a  vector  |(  to  the  normal,  i.e.  ±  to  dp,  the  equation  of 
the  tangent  may  be  written 

(<r-p)\v  =  0, (265) 

and  that  of  the  normal, 

(o--p)v  =  0. (266) 

82.  77ie  circle.    The  equation 

P  =  a(iicos0  +  i2sin^) (267) 

represents  a  circle  whose  radius  is  a ;  for,  taking  the  co-square, 
we  have 

p?  =  a-(cos^  6  +  sin^  6)  =  ar,  or  Tp  =  a, 

which  is  thd  scalar  form  of  the  equation,  and  evidently  belongs 
to  a  circle  of  radius  a,  with  the  origin  at  the  center. 
If  Cc  be  the  center,  and  the  origin  be  at  e^,  let 

e,  —  Co  =  e  and  p  —  eo  =  /o ; 
then  the  equatioil  of  the  circle  may  be  written 

T{p-e,)  =  T(p-e)  =  a, (268) 

or,  squaring  and  transposing, 


f^-2p\€  =  a' 


or,  again,        p\{2e-p) 


=  .-a4 (2«») 


98  DIRECTIONAL  CALCULUS.  [Art.  83. 

In  the  last  form  the  equation  gives  an  immediate  proof  of 
the  proposition  that  the  product  of  the  segments  of  a  secant 
line  through  a  given  point  is  constant,  and  equal  to  the  square 
of  the  tangent  from  the  point.  This  may  be  easily  seen  by 
drawing  a  diagram.  If  a^  =  ^,  the  origin  is  at  a  point  of  the 
curve,  and  the  equation  becomes 

p|(2e-p)  =  0, (270) 

which  shows  that  the  angle  inscribed  in  a  semicircle  is  a  right 
angle. 

If  we  have  two  circles  whose  equations  are 

(p  —  ci)-  —  ttf  =  0  and  (p  —  €2)-  —  ai  =  0, 
then  the  equation 

{p  -  ci)-  -  «i'  =  (p  -  £2)-  -  a-2 

is  that  of  some  curve  passing  through  the  common  points  of 
the  two  circles.  The  first  member  of  the  equation  is  the 
square  pf  the  distance  from  the  end  of  p  to  the  point  of  con- 
tact of  a  tangent  to  the  first  circle  drawn  through  the  end 
of  p,  while  the  second  member  has  a  corresponding  meaning 
for  the  other  circle :  hence  the  equation  is  the  locus  of  points 
from  which  equal  tangents  can  be  drawn  to  the  two  circles. 
Expanding,  it  reduces  to 

2p,\(e,-e,)  =  ei-a,'-e,^  +  a,', (271) 

a  straight  line  called  the  axis  radical. 

83.  Exercises. —  (1)  Show  that  the  equations 

f^  =  k(a\p-\-C)  and  f^  =  Jc'(ap  +  C') 

represent  circles,  and  find  their  radii,  and  the  vectors  to  their 
centers.  Also,  if  C=  C  =  0,  show  that  the  two  circles  cut 
each  other  orthogonally,     (a  is  some  constant  vector.) 

(2)  Show  that  the  three  axes  radical  of  three  circles  have  a 
common  point. 


Chap.  III.]       APPLICATIONS  TO  PLANE  GEOMETRY.  99 

(3)  Show  that  if  cj,  cg,  £3  are  three  vectors  drawn  outward 
from  a  common  point,  and  they  are  connected  by  the  relation 

«i«2  •  £3-  +  «2C3  • «!-  +  csci  •  €2-  =  0, 
then  their  outer  ends  lie  on  a  circle  through  their  common 
point. 

(4)  By  eq.  (97)  and  the  relation  given  in  the  last  exercise 
show  that  the  equations 


£1  €2        €2  C3        £3  £1 
22  22  22 

Cl-  «2-  ^2-  «3-  «3-  «1- 


£l£2  EoEs  £3£l 

also  hold  between  three  vectors  which,  being  drawn  outward 
from  a  point,  terminate  in  a  circle  passing  through  this  point. 

(5)  If  perpendiculars  be  drawn  from  a  point  upon  the  three 
sides  of  a  triangle,  and  the  feet  of  these  perpendiculars  be  col- 
linear,  then  will  the  locus  of  the  point  be  a  circle  circumscribed 
about  the  triangle. 

(6)  Show  that  the  tangent  line  to  the  circle  of  eq.  (267) 
has  the  equation 

o-  =  a(ii  cos  6  +  i2  sin  $)  +  ua(i2  gos6  —  ii  sin  0) , 

of  which  the  scalar  form  is  a\p  =  a^.     Also  the  equations  of 
the  tangent  and  normal  to  (269)  are  respectively 

(o-  —  €)\(p  —  e)  =  a?  and  (o-  —  £) (p  —  e)  =  0. 

(7)  Find  what  the  equation  o-|/3  =  a?  represents  when  p  is 
not  the  vector  to  a  point  on  the  circle. 

84.    The  parabola.    The  equations 

T7  +  ^n (272) 

represent  a  parabola ;  for,  eliminating  x,  we  have 

/>  =  £ti  +  2/'2,  .     .  -V^ (273) 


100  DIRECTIONAL   CALCULUS.  [Art.  84. 

which  shows  that  the  abscissa  varies  as  the  square  of  the  ordi- 
nate, a  property  of  the  parabola. 
Differentiating  (273),  we  have 

dy     2  a 

a  vector  parallel  to  the  tangent  at  the  end  of  p;  hence  the 
equation  of  the  tangent  may  be  written 

P  =  fh  +  yh  +  {fh  +  <-X (274) 

4a  \2a  J 

in  which  y  is  to  be  taken  as  constant. 

Eliminating  y  from  (273),  we  have  the  scalar  form  of  the 
equation,  viz. : 

(ph)^  =  4a.pK, (275) 

or,  as  it  may  be  written, 

p|(i2-pli2-4aii)  =  0. 

In  this  latter  form  we  see  that  the  vector  ij  ■  pjig  —  4  aii  is 
always  perpendicular  to  p.     Let 

o-  =  t2  •  /3|t2  —  4  ail ;  then  or|ii  =  —  4  a, 

and  it  appears  that  the  locus  of  the  end  of  o-  drawn  outward 
from  the  origin  is  a  right  line  parallel  to  i^,  at  a  distance  of  4  a 
to  the  left  of  the  origin;  also,  o-|t2  =  p|t2>  so  that  the  projec- 
tions of  p  and  o-  on  i2  are  equal.  The  following  proposition  is 
a  consequence,  viz. :  If  a  right-angle  triangle  have  its  rectan- 
gular vertex  fixed,  and  one  of  the  other  vertices  moves  on  a 
right  line  to  which  the  hypotenuse  remains  perpendicular,  then 
the  third  vertex  generates  a  parabola. 

To  find  the  locus  of  the  middle  points  of  a  system  of  parallel 
chords,  i.e.  a  diameter.  Let  c  be  parallel  to  the  chords,  and 
let  the  equation  of  some  chord  be  p  =  p^-\-  xt.,  in  which  pi 
satisfies  eq.  (275).  Substitute  this  value  of  p  in  (275)  to  find 
the  other  end  of  the  chord ;  therefore 

((/oi  -^  a;e)|i2)^  =  4a(pi  -f  a;e)|ti, 


Chap.  III.]       APPLICATIONS   TO  PLANE   GEOMETRY.  101 

or  (pihY  +  2x€\l2  •  p|i2  +  ^{^IhY  =  4api|ti  +  4atcc|ii; 

whence,  by  (275), 

4ae|ti  — 2£|t2-pi|t2 

If  <r  be  the  vector  to  the  middle  point  of  the  chord,  then 

1      1  I     2tte|ti  —  £  lo  •  Di  to 

whence        o-|i2  =  2  a  ^, 

which  is  the  equation  of  a  right  line  parallel  to  ij.  Hence  the 
diameters  of  a  parabola  are  all  parallel.  If  e  =  i2,  o-|t2  =  0,  so 
that  the  line  through  the  origin  parallel  to  ij  bisects  a  system 
of  chords  perpendicular  to  it.  This  line  is  the  axis  of  the 
parabola.    . 

85.  Exercises.  —  (1)  Show  that  T(p  —  aii)  =  a  +  p\ti,  and 
interpret  the  equation. 
We  have,  by  (275), 

(pIl^Y  =  p^-  {p\ciY  =  4aplti  +  0?-  a\ 
or  p-  —  2ap\ii  +  a^  =  {p\iiY  +  2a/3|ti  +  a?, 

whence        (p  —  aij)-  =  (a  +  plii)^ 
The  interpretation  is  easily  obtained  by  a  figure. 

(2)  Show  that  dp{ii+U{p  —  ai^)  =  0,  and  interpret  the 
equation. 

(3)  Show  that  the  equations  to  the  tangent  and  normal  to 
(276)  may  be  written  respectively 

o-|i2  •  plt2  =  2  a  (o-  +  p)  |li 

and  (r|ii  '^112  =  pI'2  •  pI'i  +  2  a  (p  —  or)  |t2. 

(4)  Show  that  p  =  ^t\  +  ^C2  and  (^ipY  +  2 qeg  •  caP  =  0,  are 
respectively  the  vector  and  scalar  forms  of  the  equation  of  a 
parabola  referred  to  a  tangent  whose  direction  is  ^  and  a 


102  ,  DIRECTIONAL  CALCULUS.  [Art.  86. 

diameter  through  the  point  of  contact  whose  direction  is 
that  of  cj. 

(5)  Show  that,  with  reference  to  the  equations  of  Ex.  4,  the 

lines  o-  =  p  +  x-^  and  o-  =  p  +  2/^2  cut  the  line  o-  =  ze^  at  equal 

distances  on  each  side  of  the  origin,  and  give  the  geometric 
interpretation. 

86.  Ellipse  and  hyperbola.     The  equations 

p  =  ail  cos  6  +  6i2  sin  6 (276) 

p  =  cui  sec  ^  +  &t2  tan  ^ (277) 

represent  respectively  an  ellipse  and  hyperbola,  in  which  a  and 
b  are  the  semi-axes,  and  0  is  the  eccentric  angle.  This  will  be 
evident  at  once  to  any  one  familiar  with  the  ordinary  Cartesian 
equations,  if  we  obtain  the  corresponding  scalar  forms.  We 
find  pill  =  a  cos  6,  p\i2  =  &  sin  0,  from  (276),  whence 


(' 


*    +m=l (278) 


From  (277)  we  have  p\ii  =  a  sec  6,  p|i2  =  b  tan  0,  whence 


P\i-i\^      fP\hV 


,      >   ,    ;  =  1 (279) 

a  J      \b  ' 

Since  p|ii  and  pli2  are  the  Cartesian  x  and  y,  the  equations  are 
evidently  those  of  the  ellipse  and  hyperbola.  The  two  equa- 
tions may  be  combined  by  using  the  double  sign ;  thus 


( 


P\^l\\fP\'2 


aJ^[T>=' (^«») 


Exercises.  —  Show  that  the  equations 

Tip  +  cc,)  =  ^  C^'  +  p\c)  and  T(p  -  c\)  =  ^-(p\c,  -  ^\ 
ClVC  y  a\^  C  J 

in  which  c  =  Va^  —  b^  and  c'  =  Va^  +  6^  are  equivalent  to 
(278)  and  (279)  respectively,  and  interpret  these  forms  of  the 
equations. 


Chap.  III.]       APPLICATIONS   TO  PLANE  GEOMETRY.  103 

Show  that  the  equations 

T(p  +  CL,)  +r(p-cii)  =2a 
and  T(p  +  c\)  -  T{p  —  c'tj)  =  2  a 

are  also  equivalent  to  (278)  and  (279)  respectively,  and  inter- 
pret them. 

Let  us  write  the  equation 

'-4^±^-*. (281) 

This  expression  is  a  linear  and  vector  function  of  p,  and,  by 
the  aid  of  (281),  equation  (280)  becomes 

p4>P  =  l (282) 

This  remarkably  simple  equation  may  represent,  as  will 
appear  hereafter,  not  merely  (280),  but  any  equation  of  the 
second  degree  in  p,  which  contains  no  first-degree  terms.  It 
may  thus  represent  not  only  any  central  conic  with  the  origin 
at  the  center,  but  also  any  central  quadric  referred  to  its  center 
as  origin,  when  we  are  dealing  with  three-dimensional  space, 
and  the  corresponding  locus  in  n-dimensional  space.  Similarly, 
if  <^p  be  a  linear,  point  function  of  a  variable  point  p,  we  shall 
see  that  p\4>p  —  ^  may  represent  any  conic  whatever  in  two- 
dimensional  space,  any  quadric  whatever  in  three-dimensional 
space,  and  any  locus  of  the  second  order  in  ji-dimensional  space. 

In  the  form  given  above  the  </>  function  will  be  found  to 
possess  the  following  properties,  viz. : 

(a)  (t>(p  +  cr)=(f,p  +  (fio- 

iP)  <l>(xp)  =  x<f>p 

(y)  d{.l>p)  =  <l>{dp) 

(8)  (T\(f>p  =  p:(f>a- 

The  first  three  properties  are  possessed  by  any  linear,  vector 
function.  When  the  last  relation  holds,  the  function  is  said 
to  be  self -conjugate,  and,  in  dealing  with  curves  and  surfaces  of 
the  second  order,  <f>  may  always  be  so  taken  that  this  relation 
exists,  i.e.  ^  may  be  taken  as  self -con  jugate. 


(283) 


104  DIRECTIONAL  CALCULUS.  [Art.  87. 

87.  Tangent  and  normal.  Differentiate  (282),  having  regard 
to  (y)  and  (8)  of  (283) ;  then 

dp\<f>p  +  p\<i>dp  =  2  dp\<f>p  =  0. 

Hence  <^p  is  a  vector  perpendicular  to  dp,  i.e.  parallel  to  the 
normal  to  (282)  at  the  end  of  p.  Therefore,  if  o-  be  a  vector 
to  any  point  of  the  tangent,  and  p  the  vector  to  the  point  of 
contact,  so  that  o-  —  p  is  parallel  to  dp,  we  have  (o-  —  p)[^p  =  0, 
or,  by  (282), 

(r\<f>p  =  1, (284) 

as  the  equation  of  the  tangent  to  the  curve. 
For  the  normal  we  have  the  equation 

(a-p)ct,p  =  0 (285) 

Since  <t>p  is  parallel  to  the  normal  at  the  end  of  p,  the  projec- 
tion of  p  on  (ftp  will  be  the  perpendicular  from  the  center  on 
the  tangent  line.     By  Art.  46  this  is 

<t>p  ■  p\(f>p        <l>p  1 


i<t>py-      {<t>py=    T<f>p 


U<l>p.     .     .     .     (286) 


Hence  t?ie  length  of  the  vector  <^p  is  the  reciprocal  of  that  of  the 
perpendicular  from  the  center  on  the  tangent. 

88.  Diameter.  The  diameter  being  the  locus  of  the  middle 
points  of  a  system  of  parallel  chords,  we  may  find  its  equation 
as  follows.     Let  the  system  of  chords  be  parallel  to  e,  and  let 

p  =  pi+X€ 

be  the  equation  of  one  of  them,  in  which  pi  is  a  vector  of  the 
curve,  i.e.  pi|<^pi  =  l.  Substitute  this  value  of  p  in  (282),  and 
we  have,  in  order  to  find  the  other  end  of  the  chord, 

(pi  +  xe) \<f>(pi  4- X€)  =  1,  or  pi|</>pi  +  2x'€]«^pi  +  x'clffx.  =  1, 

whence,  because  of  the  condition  above, 

x  =  0,  and  x  =  —  "  ,  ,     • 
e|9c 


Chap.  III.]       APPLICATIONS   TO   PLANE   GEOMETRY.  105 

Now,  if  o-  is  the  vector  to  the  middle  point  of  the  chord,  we 
have 

c\<f>pi 

cr  =  pi  +  ixc  =  pi--^.e. 

Multiply  into  \<f>e,  and  we  have 

<r|<^e  =  0, (287) 

an  equation  independent  of  pj,  and  depending  only  on  the  given 
direction  c  and  the  function  <^.  (287)  is  therefore  the  equation 
of  the  required  locus,  which  is  a  straight  line  perpendicular  to 
<f>€,  and  consequently  parallel  to  the  tangents  to  the  curve  at 
the  ends  of  a  diameter  parallel  to  e.  The  direction  of  <r  is  said 
to  be  conjugate  to  that  of  «,  and  the  diameters  parallel  to  c  and 
o-  are  conjugate  diameters. 

If  a  and  /3  are  any  two  conjugate  vector  semi-diameters,  they 
must  be  subject,  therefore,  to  the  conditions 

al^=^I^^  =  l. 

a|«/»/8  =  /8I«^=0J  ^       ^ 

The  results  of  Arts.  87  and  88  have  been  obtained  with  the 
functional  symbol  <{>,  without  any  reference  to  the  fortn  of  the 
function,  the  only  restriction  being  that  it  shall  be  subject  to 
the  conditions  (283)  ;  hence  these  results  are  general,  and  hold 
for  any  form  of  the  linear  vector,  self-conjugate  function. 

89.  Further  development  of  the  ff>  function.     Write 

<i>p  =  gih'p\<-i  +  g-^-2'p\h, (289) 

so  that,  comparing  with  (281),  we  have 

Px  =  ^,    g.=  ±l (290) 

Putting,  in  (289),  successively  tj  and  ij  for  p,  we  find 

<^ti  =  grill  and  «^i2  =  9'2i2 •     •     (291) 

Next  substitute  ^p  for  p  in  (289)  ;  therefore 

=  S'lti  •  p\^i-\  +  9^1  •  p\^h  =  9'Ai  •  pVi  +  9ih  •  p\h' 


106  DIRECTIONAL   CALCULUS.  [Art.  89. 

We  have  the  fourth  member  from  the  third  because 
^p|ti  =  ii\<ftp  =  pI^ii, 
by  (79)  and  (283). 

Similarly,       'i>{<i>'^p)  =  <^V  =  Qih  •  pYi  +  gih .  p\i2,  etc. ; 
so  that,  if  n  is  a  positive  whole  number, 

<^V  =  g'i%  .  pK  +  9'2%  •  pk2 (292) 

Let  m  be  some  other  positive  whole  number ;  then  also 

<^>  =  gi'h  '  p\h  +  9 "^2 '  p\h ; 
hence  ^"<^>  =  ^""^>  =  9i\  .  ii!^>  +  g2"h  •  la!^"/) 

^9'i"+%-p!'i  +  9'2"-^%-p!t2.      .     .     (293) 
Suppose  m  to  be  negative  and  equal  to  —  w ;  then  by  (293) 

^"  (^"»  =  ^V  =  h  •  p\h  +  H  •  9\h.  =  9, 

so  that  the  negative  exponent  gives  a  function,  such  that  the 
operation  indicated  by  ^  with  a  positive  exponent  of  the  same 
numerical  value  being  performed  upon  it,  gives  p  as  a  result ; 
i.e.  the  operations  ^^  and  <^~"  cancel  each  other.  Hence  (292) 
and  (293)  hold  both  for  positive  and  negative  values  of  the 
exponents. 

Finally,  suppose  n  =  — ;  then  we  ought  to  have 

^      «»j       {<f>^ip)  =  <l>p, 

if  the  exponential  law  holds  for  this  case,  and  the  result  is 
easily  verified  as  before.  Thus  (292)  holds  for  all  real  values 
of  n. 

If  f(x)  =  Ax""  +  5x"-i  +  . .  •  ^, 

we  may  easily  show  that 

^P  =  (M))p=f(90-ii'pW+f(92)-h'p\h,    .     (294) 

in  which  4»  =f(^<f>)  is  also  a  linear,  vector,  self -con  jugate  func- 
tion. 


Chap.  III.]       APPLICATIONS  TO  PLANE  GEOMETRY.  107 

Again,  if /^  and^  are  functional  symbols  of  the  same  form 
as /above,  let  *  =47TT  =  ^i^2~^ ;  then 

*"=*'*''"  =Ito)  •"•"''■  +/^teJ  •"• "''» •  (2**^^ 

so  that  ^  is  still  a  function  of  the  same  kind  as  <;^. 

Finally,  let  <^,  <^',  <^"  be  three  functions  of  the  form  (289) 
with  corresponding  g^,  g^,  g^",  g^,  gi,  g^' ;  then 

=  {Jcg,+k'g,'-{-k%"),, .  p|4  (• .      (296) 

+  {kg,-\-k%' +k"g,")  i^  •  p\i^  =^p     ^ 

All  the  results  of  this  article  hold  equally  well  for  w-dimen- 
sional  space  when  <f>p  =  ^"(g  •  f  p|t). 

90.  Exercises.  —  (1)  Show  that 

*»'  +  iV  =  |(i:  +  i)'.-p|..±|,(i  +  i).-.|v 

(2)  Show  that  ti!«^(<^-  - 1)  "'ti  =  t^—* 

(3)  Show  that  (^^  +  ^y  =  2(^)p. 

(4)  Show  that 

<^(«/,2-l)->(<^+3)-V  =  i(<^  +  l)-^+i(<)^-l)-\-f(<^+3)-'t, 


(5)  Show  that,  if  <f>  and  ^'  are  of  the  form  (289), 

ffxfi'p  =  <ti'^p. 

(6)  Show  that 
(<^''-2«/,+l)p=(<^-l)%and(</,Hl)(<^+l)-V  =  (<^'-<^+l)/>. 


108  DIRECTIONAL  CALCULUS.  [Art.  91. 

91.    The  function  <f>K     Since  </>  =  <f>^<fi^,  we  have 

whence,   T<^^p  =  1, (297) 

which  is  a  form  of  the  equation  of  the  central  conic  analogous 
to  that  of  the  circle.    We  have 

4>  P  =  9i  1-1  •  pi'i  +  92^'-2 '  p\'-2  =  -•'!•  ph  +  ,     •  '2  •  pK 

a  "V  ±  1 

so  that,  in  the  case  of  the  hyperbola,  <^^  is  an  imaginary  func- 
tion; nevertheless  (297)  is  real  and  equivalent  to  (279),  as  is 
easily  seen. 

Let  a  and  ^  be  conjugate  vector  semi-diameters ;  then 

a|«^;3  =  0=a|«^^^/8=<^^/3|<)!)^a; (298) 

hence  <f>^a  and  (^-yS  are  unit  normal  vectors.  We  will  determine 
what  relation  ^*p  bears  to  p  in  the  ellipse. 

be  the  equations  of  two  ellipses  having  the  a  axis  in  common, 
and  let  p  and  p'  be  so  taken  that  they  have  the  same  projec- 
tion on  ij ;  that  is,  p\li  =  p'|ii. 

Then       <I>'Wp  =  (^ '  p'I'i  +  '§,  ■  p'\^^(^^  '  p\^i  + 1 '  P^^ 

_  p'lh  '  p\i-2  _  p'lh-  p\h  _  pK  fph  _  p'\h\ 
ah  ab'  a\b        b' )' 

But;  from  the  equations  of  the  curves, 
hence  <t>'^p'<f>^p  =  0. 


1      iP'\'-A'  —  (P'\^2'^ 
a  Kb' 


Chap.  III.]       APPLICATIONS   TO   PLANE   GEOMETRY.  109 

Thus  it  appears  that,  if  any  two  ellipses  have  a  common 
axis,  and  p  be  taken  in  each  so  as  to  have  the  same  projection 
on  this  axis,  then  (f>^p  will  be  the  same  unit  vector  for  each 
ellipse.  Let  one  of  the  ellipses  be  a  circle  of  radius  a,  i.e.  let 
6'  =  a ;  then  a^</)'p'  =  a(f>'^p'  =  p',  and  p'<f>y  =  0,  so  that  <fi^p  in 
any  ellipse  is  a  unit  vector  laid  off  along  that  radius  of  the  circle, 
described  on  either  axis,  which  has  the  same  projection  on  this 
axis  that  p  has.  By  (298)  it  appears  that  the  radii  of  this 
circle  corresponding  to  a  pair  of  conjugate  semi-diameters  are 
mutually  perpendicular. 

92.  Interpretation  of  the  equation  (r\<f>€  =  1.  If  c  satisfies  the 
equation  e\<t>€  =  '[,  it  is  a  vector  of  the  curve,  and  the  given 
equation  is  identical  with  (284),  and  therefore  represents  a 
tangent  to  (282). 

If  this  is  not  the  case,  let  c  be  first  a  vector  to  some  point 
from  which  a  tangent  can  be  drawn  to  the  curve.  Let  pi  and 
P2  be  vectors  to  the  points  of  contact  of  the  two  tangents  which 
pass  through  the  end  of  c.    The  equations  of  these  tangents  are 

o-[0pj  =  1  and  cr|<^p2  ^  1  > 

and  since  they  pass  through  the  end  of  e,  the  equations  must 
be  satisfied  when  e  is  substituted  for  a-.  Thus  we  have  the 
relations 

«1M  =  1  =  c|^/)2) 

which  must  always  hold  between  c,  pi,  and  pi. 

If  now,  in  the  given  equation  <T|</)e  =  1,  we  make  o-  =  pi,  or 
o-  =  p2,  it  appears  that  the  equation  is  satisfied,  and  hence  the 
line  represented  by  it  passes  through  the  points  of  contact  of  the 
tangents  through  the  end  of  c.  Furthermore,  the  line  is  perpen- 
dicular to  ^e,  and  therefore  parallel  to  the  diameter  conjugate 
to  c. 

Next,  write  the  equation  £|0(r=  1;  one  value  of  o-  will  evi- 
dently coincide  in  direction  with  c ;  when  it  has  this  direction, 
suppose  it  to  become  fixed,  and  c  to  vary ;  the  equation  still 
represents  a  right  line,  which  will  evidently  be  parallel  to  the 


110  DIRECTIONAL  CALCULUS.  [Art.  93. 

previous  one ;  and  since  one  value  of  e  will  be  its  original  value, 
this  line  must  pass  through  the  end  of  the  original  e. 
The  line  represented  by  the  equation 

o-l<^e=l (299) 

is  the  polar  of  the  point  at  the  end  of  c. 

Let  ci  and  cj  be  the  vectors  from  the  origin  to  two  points  p^ 
and  P2 ;  then,  if  the  polar  of  pi  passes  through  p^,  that  of  Pi  will 
also  pass  through  py.  For  the  equations  of  the  polars  of  pj  and 
P2  are  respectively  o-|<^ei  =  1  and  o-]<^e2  =  1 ;  if  the  first  passes 
through  P2,  we  must  have  £2l^«i  =  ^  >  but  this  is  also  the  condi- 
tion that  the  second  shall  pass  through  pi.  Since  p.,  may  be 
taken  at  any  point  of  the  polar  of  pi,  while  Us  polar  always 
passes  through  pi,  it  appears  that  if  a  point  move  along  a  fixed 
right  line,  its  polar  passes  through  a  fixed  point ;  and  recipro- 
cally, if  a  line  passes  always  through  a  fixed  point,  its  pole 
moves  on  a  fixed  right  line. 

The  semi-diameter  along  c  is  a  mean  proportional  between  Te 
and  the  distance  along  c  from  the  center  to  the  polar  of  e.  Let 
the  semi-diameter  be  xe,  and  the  distance  to  the  polar  yc ;  then 
we  must  show  that  Tt-yTi  —  {xT^y,  or  a?  =  y.  Put  X€  for  p 
in  (282),  therefore  a;^e|<^£  =  1 ;  also  put  ye.  for  o-  in  (299),  and 
we  have  yt\^(.  —  l;  therefore  a?  =  y.  q.e.d. 

93.  Exercises.  —  (1)  Show  that  the  system  of  conies  ob- 
tained by  giving  different  values  to  k  in  the  equation 

p|(<A-'-A:)-V  =  l 

have  all  the  same  foci.     These  are  called  confocal  conies. 

(2)  Through  any  point  in  the  plane  of  the  confocal  system 
there  pass  two  conies  of  the  system,  which  cut  each  other  at 
right  angles. 

Let  e  be  the  vector  to  any  point;  then,  if  the  curve  pass 
through  it,  we  must  have  the  equation  of  ex.  (2)  satisfied  when 
e  is  substituted  for  p.    Hence 

el(<^     -fc)     c-l-^^-^-f-p— ^, 


Chap.  III.]       APPLICATIONS   TO   PLANE   GEOMETRY.  Ill 

a  quadratic  in  k,  giving  two  values  corresponding  to  the  two 
curves  through  the  end  of  e.  Let  the  roots  of  this  equation  be 
ki  and  k2 ;  then  the  curves  are 

p\{<f>-'  -  A-,)  V  =  1  and  pi(«^-'  -  k,y'p  =  1, 

and  (^~^  — A;i)~'e  and  (^~^  — fca)"^^  are  vectors  parallel  to  the 
respective  normals  at  e,  whose  co-product  must  be  zero  if  the 
normals  are  at  right  angles.     Hence 

(<{>-'  -k,)-h\{cl>-'  -  h)~h  =  c|(«^-i  -  k,)-\cf>-'  -  k,)-h 


A/2  ^~  ^1  1^2  ~~'  *^1 

(3)  If  pi  and  p2  are  any  two  vectors  of  the  central  conic, 
show  that  p2  —  pi  and  p^  -f  pi  (supplementary  chords)  are  con- 
jugate in  direction. 

(4)  Show  that  the  equations  of  the  diagonals  of  the  paral- 
lelogram formed  by  the  tangents  at  the  ends  of  pi,  p2,  —  pi, 
and  —  p.2  are  o-|^(p2  — pi)  =  0  and  (t\4>{pi-\-  p2)  =  ^,  and  that 
these  diagonals  are  conjugate  in  direction. 

(5)  Find  the  condition  that  the  line  or|c  =  C  shall  be  tangent 
to  the  curve  p|^p  =  1. 

The  equation  of  the  tangent  is   (r|<^p  =  1 ;   comparing  this 

with  the  given  equation,  we  have  ^p  =  — ,  or  p  =  — — .     Sub- 
stituting  this  value  of  p  in  the  equation  of  the  curve,  we  have 
=  1,  or  C^  =  c|^"^c,  the  required  condition.    Thus  the 


<t,-\\€  _ 


c 

line  whose  equation  is 

(r|c  =  V£|<^-^c=  Tcf>-^e (300) 

is  always  tangent  to  the  central  conic. 

94.    TJie  conic  referred  to  conjugate  diameters. 
The  equation 

ipay±{pfty=iafiy (30i) 


112  DIRECTIONAL  CALCULUS.  [Art.  95. 

represents  an  ellipse  or  hyperbola  referred  to  the  conjugate 
semi-diameters  a  and  (3.  The  equation  is  satisfied  when  p  =  (3, 
so  that  the  curve  passes  through  the  end  of  (3 ;  also,  the  curve 
is  satisfied  when  p  =  aV±l,  so  that  the  end  of  a  is  a  real 
point  of  the  ellipse,  and  an  imaginary  point  of  the  hyperbola. 

so  that  (301)  becomes  p\<f>p  =  l;  then  we  have 

and  a  and  fi  are  conjugate  in  direction,  by  (288).  Equation 
(301)  shows  that,  in  the  hyperbola,  if  any  diameter  cuts  the 
curve,  its  conjugate  does  not. 

Exercise.  —  Show  that  the  ellipse  and  hyperbola  of  (301) 
are  also  represented  by 

p  =  a  cos  ^  +  y8  sin  ^  and  p  =  a  sec  ^  -f  ^  tan  6, 

respectively. 

95.  TJie  area  of  the  parallelogram  formed  by  tangents  at  the 
ends  of  conjugate  diameters  of  an  ellipse  is  constant. 

If  a  and  /8  are  any  pair  of  conjugate  vector  semi-diameters, 
then  the  area  is  4o)8.    Now 

a  =  <}>~^<f>*a  =  ail  •  ii\(f)^a  +  l>h  '  h\*l>  c^} 

hence  4a)3  =  4a6*'''^,'^   ''''^■"  =  4 aZ>  •  i,i.,  •  <^ W^yS  =  4 a6. 

See  (298)  and  (193). 

96.  Exercises.  —  (1)  Find  the  locus  of  the  intersection  of 
tangents  at  the  ends  of  conjugate  diameters. 

Taking  a  and  /8  as  usual,  we  have  o-  =  a  +  yS  =  vector  of  point 
whose  locus  is  required.     Therefore 

a|<^<7=(a+)S)|<^(a+/8)=aI«^+2al<^/8+i8>)8=l-t-0  +  l  =  2, 


Chap.  IU.]      APPLICATIONS  TO  PLANE  GEO^SIETRY.  113 

a  similar  curve,  which  is,  in  this  case,  an  ellipse.  If  the  given 
curve  be  an  hyperbola,  the  locus  of  (r=a+f3  becomes  o-]«/)o-=0, 
the  asymptotes  of  the  curve,  because,  in  this  case,  if  fi\<f>^  =  1, 
a|^  =  —  1. 

(2)  Find  the  locus  of  the  extremity  of  (f>p ;  also  of  ^"p,  not- 
ing particularly  the  case  when  7i  =  ^. 

(3)  Show  that,  when  T<j>p  =  c,  the  locus  of  p  is  the  curve 
whose  equation  is  pI^V  =  ^• 

(4)  Show  that  the  locus  of  the  foot  of  the  perpendicular 
drawn  from  any  point  to  the  tangent  to  the  conic,  i.e.  the  pedal 
curve,  has  the  equation  (o-  —  c)|<^~^(or  —  c)  =  [or|((r  —  e)]^ 

(5)  If  cj  and  c,  are  any  two  unit  vectors  at  right  angles  to 
each  other,  show  that  ei!<^"'ei  +  e2\4>~^€2=  a^  ±  b\ 

The  conditions  give  c,  =  l^i-     Thus  we  have 

and  e,\<l>-'c,  =  \e,\i>-%  =  a\c,i,y  ±  6^(^12)' ; 

whence,  adding,  we  have  the  result. 

(6)  If  a  and  /8  are  vector  conjugate  semi-diameters,  show, 
by  the  result  of  the  last  example,  that  a-  +  ^  =  a^  ±  b^. 

By  Art.  91  <^V  and  <f>-p  are  unit  normal  vectors ;  real,  for  the 
ellipse,  and  imaginary,  for  the  hyperbola.    Hence 

(/)^a|<^-»<^^a  +  <^^^|<^-V-/3  =  a"  ±  62  =  a?  +  ^. 

(7)  Show  that  the  locus  of  the  common  point  of  perpendic- 
ular tangents  is  a^  =  a-  ±  b-. 

Use  eq.  (300). 

97.  Tlie  general  linear,  vector  function  in  plane  spaxx.  The 
most  general  form  of  this  function  may  be  written 

<j>p  =  €1 '  €i'\p -{■  €2 '  e^'lp  : (302) 

for,  suppose  there  were  other  terms  such  as  eg  •  cg'lp  +  kp ;  write 

cg' =  mic/ +  ?7i2«2'   and    p  = -— (|ci' -caV  -  K*  CiV), 

C1C2' 


114  DIRECTIONAL  CALCULUS.  [Art.  98. 

and  these  terms  become 

1^      \   i\     III/  1     "'      1(1     If 

W1C3 —  .  Ica'  J  •  p\ei  +  (  WI2C3  +  —r-,  •  i«i   1  •  P\^2 , 

which,  on  being  combined  with  the  terms  composing  (302), 
give  an  expression  of  the  same  form  as  before,  merely  having 
instead  of  e^  and  ca  the  vectors 

£1  +  wijcg —  •  lea'  and  cj  +  mot^  +  -— -  •  \t^. 

ei  £2  £1  £2 

This  general  form  of  </»  possesses  all  the  properties  given  in 
(283)  except  the  last,  or  self-conjugate  property.  If,  how- 
ever, we  write 

<i>cP  =  ^i'  •  £i!p  +  £2'  •  €2\p, (303) 

then  we  shall  have 

p|^o-  =  (r|<^eP  and  pI^.o- =  o-|<^/3 (304) 

0c  is  called  the  conjugate  function  to  0,  and  vice  versd. 
When  0,,  =  «^,  the  function  is  self-conjugate,  and  the  condition 
(8)  of  (283)  is  satisfied.  In  all  geometrical  applications  it  is 
possible  to  take  ^  self -con  jugate,  and  we  shall  always  so  regard 
it,  because  it  greatly  facilitates  necessary  operations. 

The  sum  of  a  linear  function  and  its  conjugate  is  always  self- 
conjugate:  for 

(r\((f>  +  <l>c) p  =  <t\<I>p  -h  <t\<}>cP  =  pHc^r  +  p\^(^  =  pX4>  +  «/>c)o'- 

98.  Inversion  of  the  <fi  function.  Suppose  we  wish  to  find 
the  value  of  p  from  the  equation 

<t>p  =  c (305) 

This  gives  p  =  <^~'£; (306) 

and  if  we  can  find  the  form  of  the  inverse  function,  we  have 
the  solution  of  (305).  For  convenience  write  e  =  |A,  so  that 
(f>p  =  \X  and  p  =  (f)~^\\ ;  then 

X\<f>p  =  0  =  p\<l>^, 

whence  p  =  x\<l)^\  =  <f>~^\\. 


Chap.  III.]       APPLICATIONS  TO   PLANE  GEOMETRY.  115 

To  determine  x,  let  /u.  be  any  vector  whatever ;  then 

...  x=     ^^     , 
and  p=<^-'e  =  </>-'|A  =  ^^;'f''^ (307) 

This  equation  affords  a  solution  of  (305),  but  we  proceed  to 
obtain  a  formula  more  convenient  than  this  for  most  purposes. 
Let  A  and  /u.  be  now  any  two  constant  vectors  whatever ;  we 
have,  from  (305), 

X\(f)p  ^  p\(j>J^  =  A.|£  and  fi\4>p  ^  p\4>cf^  =  /*!'  5 

also,  writing  p  in  terms  of  its  projections  on  \^^X  and  \<f)cfi,  we 
have 

or,  substituting  from  the  preceding  equations, 

r'^  =  p  =  i:^(\^^-t-\^-\^ci^'M^),  •   (308) 

<pA9cP- 

the  proposed  formula.    If  <^  be  self-conjugate,  then  ^„  =  </»,  and 
the  suffix  may  be  dropped  in  (307)  and  (308). 

To  invert  such  a  function  as  <;^i  +  Jc<f>2  +  A;'</>3  +  etc.  =  $,  say, 
we  have  4>„  =  <^i  +  A;<^2  +  ^■'<^3  +  etc.,  and  4>  and  $„  are  to  be 
substituted  for  </>  and  </>,  in  (308).  For  example,  suppose 
^=  cf)  -\-  g  ;  then 

/,    ,  ,A-i        I  (<^o  +  g)\  -p-lp-l  (<^c  +  9)fi '  X.\p 
^"^^'^    ^=  ii>.  +  9)X(cf.^  +  9)p^ 

_        ^~V  •  ^c^^cP-  +  ^p-'  p-g 


mocf>-^p -\- gp 


if  we  write 


wio  +  Wigr  +  g'a 


mo  =  ^4^andm,  =  ^'^-^~^^-\   •     .     (309) 


116  DIRECTIONAL  CALCULUS.  [Art.  98. 

Hence      (wio  +  mig  +  f)  p  =  (<!>  +  g)  (mo<f>-^p  +  gp) 

=  wioP  +  9(f>p  +  mog<t>-^p  +  g-p. 
.:  m^p  =  g4>p  +  mfg<f>-^p, 

(mi  —  <f>)p 
«>'  <^  V  = ^^ (310) 

Thus  we  have  still  another  inversion  formula,  which  is,  how- 
ever, not  generally  so  convenient  as  (308). 
Operating  by  </>,  (310)  assumes  the  form 

(<t>'-m,4>  +  mo)p  =  0, (311) 

The  quantities  mo  and  ?)ii  are  invariants,  i.e.  their  value  will 
be  unchanged  if  other  vectors  be  substituted  for  X  and  p,.  For, 
let  XjX  +  2/i/x,  and  x^  +  y^  be  any  other  two  vectors  in  the 
plane  space  under  consideration,  and  substitute  them  for  \ 
and  p.: 

_  <l>,(xi\  +  yip.)<t>,(x2k  +  yop-) 
"  *''""      (aJjX  +  2/ifi)  (ajgX  +  ?/2/a) 

_  a;i.y2<^cA.<^,/u,  +  x.jyi(f>,p.(f>,X  _  <^<A<^eM_ 
x^y^p.  +  x-Tyip-k  X.p. 

Similarly, 

(a;iX  +  yip.)(f>,{x^  +  y.,p.)  —  (x.A  +  y.2p.)<t>A^'i>^  +  !/\>J-) 
^^  ~  (a^iX  +  t/]/i)  {x.A  +  y-2l^) 

_  (%V2  —  a^i)  (^<^c/^  —  /^^<A)  _  X<^,/t  —  /A<^.X 
~"  {xiy^  —  X2yi)Xp.  ~         Xp. 

From  the  above  it  follows  that,  in  any  given  case,  Ave  may 
assume  such  values  of  X  and  p.  as  may  be  most  convenient. 
For  example,  take  ^p  =  cj  •  cj'jp  +  €2  •  cj'lp?  s-s  in  (302)  ;  let  X  =  [cj 
and  /i  =  Ic, ;  then 

^A  =  ^cltl  =  «!>'  •  «ie2>     ^c/A  =  ^c'e2  =  —  «i'  •  «lC2- 

.     j:-i  ka' •  eif2- «2P+i«i'-«ie2- «iP       «iP  •  i«i' +  CaP  *  !«2' 

ei«2  •  («i«2)  eiC2'«ie2 


Chap.  III.]       APPLICATIONS   TO   PLANE  GEO:METflY.  117 

99.  Exercises.  —  (1)  Invert 

Find  p  as  a  function  of  c  in  the  following  cases  : 

(2)  a-ap-hl3-fi\p  =  €. 

(3)  Ap  +  a-a\p  =  €,  A  being  a  scalar  constant. 

(4)  Ap  +  a-ap-^ft-(3p  =  €. 

100.  The  general  equation  of  the  second  degree  in  plane  space. 
This  equation  may  be  written 

p\<t.p-{-2y\p=C. (312) 

For  every  term  of  the  second  degree  in  p  may  be  expressed 
in  the  form  p|c  •  pic',  so  that  the  portion  of  the  left-hand  mem- 
ber which  is  of  the  second  degree  will  consist  of  the  sum  of  a 
series  of  such  terms,  i.e.  %  {p\€  •  p\c') .  This  form  evidently 
includes  such  forms  as  (p|£)^  equivalent  to  p|c'p|e,  and  Af^, 
equivalent  to  A({p\iiy -^'{pli^y -\-{p\isy).  If  we  write  now 
5"(c  •  pIc')  =  <j>p,  the  second-degree  terms  assume  the  form  p\<j>p, 
but  ^  will  not  be,  in  general,  self-conjugate.  If,  however,  we 
write 

i2';(e.ple'-f-c'.p|c)=<^p, (313) 

then  the  second-degree  terms  will  still  assume  the  form  p\<f)p, 
and  <^  icill  be  self-conjugate  because  it  is  the  sum  of  a  linear 
vector  function  and  its  conjugate.     See  Art.  97. 

The  self-conjugate  form  of  <^  for  any  set  of  second-degree 
terms  may  be  obtained  by  differentiating  these  terms,  a  method 
which  may  be  convenient  for  the  beginner.  For  suppose  <f>  not 
self -conjugate  in  the  expression  pj^p  ;  differentiating,  we  have 

dp|<^p  +  p\<fidp  =  dp\<i>p  4-  dp\4>^  =  dp\{^  +  <l>,)p, 

and  <ti  +  <i>c  is  self-conjugate.     Also, 

p1<^./>  =  pl't^Pi  so  that  ip\  (<f>  -\-  <t>^)p  =  pl<^p. 


118  DIRECTIONAL   CALCULUS.  [Art.  lOL 

For  example,  let 

p\<l>p  =  p--i-p€-  p\e'. 

Then,  2dp\p+dp€  •  pl^'+pe  ■  dp\t'  =  dp\{2p-\c  -  pje'+e'  •  pe) 

is  the  differential ;  and  if  we  put  </)p  =  p  -f-  ^^(e'  •  pe  —  je  •  pie'), 
we  have  p|<^p  =  p-  +'pe  •  p\e'.  All  terms  of  the  first  degree  may 
evidently  be  combined  in  one  ;  for  example, 

tiP  +  ealp  +  talp  +  etc.  =  p\{\ei-\-e.2-\-e.,)  =  p\(2  y)  =  2  y\p, 

so  that     y  =  i(|£i+eo  +  £3)- 

101.  To  determine  the  locus  represented  by  eq.  312,  we  will 
first  change  the  origin  to  the  center,  if  the  curve  has  such  a 
point.  Let  S  be  the  vector  to  the  centei-,  and  p'  the  new  vector 
radius  of  any  point;  then,  p  =  S  +  p'.  Substituting  in  (312), 
we  have 

(S  +  p')\<l>i8  +  p')  +  2y\{8  +  p')=C, 

or,  expanding, 

8\<i>S  +  28|  V  +  p'l  V  +  2yj8  +  2y|p'  =  a 

If  the  origin  be  now  at  the  center,  the  terms  of  the  first 
degree  must  disappear,  because  the  equation  must  be  unchanged 
when  —  p'  is  substituted  for  +  p'.     Thus  we  must  have 

p'|(<^S  +  y)  =  0. 
But  this  is  to  be  true  for  all  values  of  p' ;  hence, 

<^8+y=0,  or  8=-r^y  =  ^1^  •  '^^-/'y  •  l'^^.      (314) 

<pA<pp. 

Eq.  314  gives  the  value  of  8,  the  vector  to  the  center.  When 
<^A</>p.  differs  from  zero,  the  locus  has  a  finite  center  ;  but  when 
<f>\(f>fx.  =  0,  the  center  is  at  oo,  unless  the  numerator  is  also  zero. 
In  this  last  case,  8  becomes  indeterminate  ;  it  is  in  fact  the 
vector  radius  to  a  straight  line,  the  locus  of  centers,  whose 
equation  is 

-«^8  =  y, ■ (315) 

and  (312)  now  represents  two  parallel  or  coincident  right 
lines. 


Chap.  III.]       APPLICATIONS  TO   PLANE  GEOMETRY.  119 

The  equation 

€f,\<l>fi  =  0 (316) 

requires  that  the  function  0  should  consist  of  a  single  constant 
vector  multiplied  by  a  scalar  factor,  or  should  be  reducible  to 
that  form.  For,  taking  the  general  form  of  <;^  as  given  in  Art. 
97,  we  have 

<t>\(f>fi  =  (ci  •  Ajei'  4-  ^2  •  Xjca')  (ci  •  /x|ci'  +  e.^  ■  //.jc.') 
=  eie,.(Xlci'.,*|e2'-X|£2'.,.|c/)=0. 

Since  X  and  /u.  are  to  have  any  values  we  choose  to  assign, 
this  equation  can  only  be  satisfied  by  making 

C1C2  =  0,  i.e.  £2  =  Hcj. 
Thus  we  have 

<^p  =  Cl.p|(£l'H-WC2'). 

But  we  are  dealing  only  with  self-conjugate  functions ;  therefore 
the  vector  appearing  in  the  scalar  factor  must  be  the  same  as 
the  other ;  i.e.  we  must  have  the  function  of  the  form 

(f>p  =  €  •  e\p •    .     .     (317) 

Hence,  when  (316)  is  satisfied,  and  the  numerator  of  the 
value  of  S  is  not  zero,  eq.  (312)  takes  the  form 

(£[/,)='  + 2  X|p=  a.     . (318) 

Since  we  know  now  that  the  center  is  at  oo,  change  the  ori- 
gin to  a  point  on  the  curve,  by  putting  p'  +  8'  for  p. 

.'.  (£l(p'  +  8'))==  +  2y\(p'  +  8')=C=  {.Ip'y  +  2£!p' .  £l8'  +  (.\B'y 

+  2y\p'-h2y\8'. 

If  the  origin  be  now  on  the  curve,  the  equation  must  be  sat- 
isfied when  p'  =  0 ;  whence 

(£l8')''-f2yi8'=C7, 

an  equation  for  determining  8'.     The  equation  thus  becomes 

(£|p')^  +  2(£.S'l£  +  y)|p'=0,  ....     (319) 

which  is  of  the  same  form  as  equations  of  Art.  84,  which  were 
shown  to  represent  parabolas. 


120  DIRECTIONAL  CALCULUS.  [Art.  102. 

102.  Resuming  now  the  general  equation  with  the  origin  at 
the  center,  it  becomes  by  (314), 

p\<f>p=C-8\<}>8-2y\8=C  +  y\<f>~'y  =  C,s3iy.  .     .     (320) 

Let  us  find  the  points  at  which  the  tangent  is  perpendicular 
to  p ;  i.e.  p  is  parallel  to  <^p,  or 

<t>p  =  gp,  or  {<t>-g)p  =  0, (321) 

g  being  a  scalar  constant  to  be  determined. 
Eqs.  (320)  and  (321)  give 

pl«^p=C'=^p2,orp2=rp  =  y'      •     •     (322) 

Multiply  the  complement  of  (321)  successively  by  any  two 
vectors  X  and  p. ;  therefore 

^\(<t>-y)p  =  p\{<f>-o)>^  =  o 

and  p.\(<l)-g)p  =  p\(<l>-g)p,  =  0. 

In  order  that  p  may  be   simultaneously  perpendicular   to 

(<f>  —  g)\  and  {<f>  —  g)p.,  these  vectors  must  be  parallel ;  hence 

we  must  have 

(<l>-g)X{<i>-g)p.  =  0 

or  g^  —  mig  -f-  mo  =  0, (323) 

Wo  and  mj  having  the  values  given  in  (309) . 

Eqs.  (322)  and  (323)  show  that  Tp  has  two  pairs  of  values, 
the  values  in  each  pair  being  numerically  equal  but  of  opposite 
sign.  Thus  there  are  four  points  at  the  opposite  ends  of  two 
diameters  at  which  p<f>p  —  0. 

Let  gi  and  g2  be  the  two  roots  of  (323),  and  pi  and  p^  the  cor- 
responding values  of  p.     Then,  by  (321),  we  must  have 

^Pi  =  9ipi  and  <^p2  =  9292- 

The  most  general  form  of  a  self -con  jugate  function  may  be 
written 

2<j>p  =  gi(€i  •  pW  +  e/  •  p\€i)  4-  92(^2 '  pW  +  ^  •  ph)' 


Chap.  III.]      APPLICATIONS   TO   PLANE  GEOIVIETRY.  121 

whence,  by  the  conditions  above, 

2«^Pl  =  9l((l  •  Plh'  +  «/  •  Pl|ei)  +  9'2(«2  •  P\h'  +  ^-2    •  Pi\^2)  =  ^9iPi, 

and 

2 ^p2  =  9'i(«i  •  P2|«i'  +  e/  •  P2|«i/  +  9'2(e2  •  pih'  +  ^2  '  P2IC2)  =  2g'2/02- 

Hence  pilco'  =  pil^z  =  ^2^1'  =  p^ki  =  ^  >  therefore  £2  and  e^  are 
both  perpendicular  to  pi,  and  hence  parallel  to  each  other, 
and  cj  and  cj'  are  both  perpendicular  to  p2,  and  hence  parallel  to 
each  other.    Therefore  let  ex'  =  C]  and  cg'  =  £2  j  then  we  have 

^Pl  =  9'l  •  «1  •  Plkl  =  9'lPl    ^^d     ^P2  =  9^2  •  C2  •  P2|f2  =  9'2P2- 

Hence  ci  is  ||  to  pi  and  cg  is  ||  to  p2,  also  ci-=  r^ei=l  =  e2-=  T^ ; 
thus  ci  and  cj  are  ^tmY  normal  vectors,  and  it  appears  that, 
whatever  may  be  the  original  form  of  <f>,  it  may  always  be 
reduced  to  the  form 

4>p  =  gii-i '  plh  +  92h  •  pK 

ii  and  12  being  unit  normal  vectors  parallel  to  the  values  of  p 
which  satisfy  the  equation  (321).     Now,  by  (322), 

C      C  .  C      C 

—.=  —.j  say,  and  g'2  =  —„ 
pi-      or  pT 


9i  =  —  =  ^,  say,  and  grg  =  —  =  — ,  say ; 

a"  Pr        Q- 


so  that  ^  has  the  form  of  (281),  and  (320)  represents  an 
ellipse  or  hyperbola,  according  as  g^  and  g^  are  both  positive, 
or  one  is  positive  and  one  negative,  provided  that  C  be  a  posi- 
tive quantity  as  it  may  always  be  taken.  If  g^  and  g^,  are  both 
negative,  the  curve  is  imaginary.  The  vectors  (^  — g'i)A.and 
(^  —  g^X,  or  {(f>  —  gri)/x  and  (^  —  g^p.  are  respectively  perpen- 
dicular to  the  axes,  which  are  thus  completely  determined. 

In  (323)  we  have  mo  =  g^g2  and  m,i  =  gx-\-g2',  hence,  when 
9 1  and  92  are  both  +,  mo  is  +  ;  and  if  one  of  them  be  — ,  Wq  is 
— ;  consequently  we  have 

+  for  the  ellipse 

0  for  the  parabola    ....     (324) 

—  for  the  hyperbola 


]  22  DIRECTIONAL   CALCULUS.  [Art.  10:^,. 

103.  Exercises.  —  (1)  To  discuss  the  equation 

In  this  case  <f)p  =  Ap  -\- nc  •  p\€ ;  suppose  that  Te  =  1,  and  put 

A  =  £  and  /x  =  |e ;  then  <f>\  =  <l>€  =  (A  +  n) e,  cftp.  =  (file  =  A  •  |c. 

.    jv  _       (A  +  n)-\€'ey-\-A€'€\y_      Ay  +  n  •  je  •  ty 
A{A  +  n)  A(A-{-n)    ' 

the  last  by  (98). 

The  center  is  at  oo,  and  the  curve  a  parabola,  first,  when 
^  =  0,  second,  when  A  +  n  =  0,  unless  at  the  same  time  the 
numerator  of  the  value  of  8  is  zero. 

71  *    €  *  CV 

When  ^  =  0,  8  = ^ — -,  and  ncy  is  zero  when  n  =  0,  Avhen 

y  =  0,  or  when  c  is  parallel  to  y.     These  cases  correspond  re- 

Ic 

spectively  to  the  line  y!p  =  W,  the  parallel  lines  e!p  =  ±-v     , 

\n 

and  the  parallel  lines  n  •  dp  =  —  m  ±  Vm^  +  nC,  in  Avhich  me 
has  been  substituted  for  y. 

When  A  =  —  n,   8  =  ^—^,  and  8  =  -,   when   v  =  t^)  or  c  is 

perpendicular  to  y.     These  cases  correspond  respectively  to 
the  pairs  of  parallel  lines 

cp  =  ± -^—  and  A-  ep  =  —  vi  ±  Vm^ -}-  AC. 

When  A(A  +  n)  is  not  zero,  the  equation  represents  a  cen- 
tral conic,  of  which  we  will  find  the  axes. 

We  find  TTio  =  A{A -f  n),  wij  =  2A  +  7i; 

hence  (323)  becomes 

g'-  (2  A+n)g+A{A+n)  =0=  (g-A)  (g-A-n) . 

Thus  a'  =  ^,  b'  =  — ^ ,  in  which  C  =  C -^  Ay'- -{- n(ey)\ 
A  A  +  n  A(A+n) 

(<f>  —  gi)\  =  ((f>  —A)e  =  nt  is  A.  to  the  a  axis,  so  that  the  axes 
are  now  completely  determined. 


Chap.  III.]       APPLICATIONS  TO   PLANE  GEOMETRY.  123 

(2)  Discuss  the  following  equations : 

(a)    (apy-±(f3pr-  =  {a{3r-. 

(6)  p^-a\p  ■  I3\p-ap  '  /3p- {a+ft)\p=l,  when  Ta=T/S=l. 

(c)  (p\(a-(3)y-(p(a+^)y=A(aPr,  when  Ta=T{3=l. 

(d)  e^p?  =  e-[c ;(p  —  e)]-,  e  being  a  scalar  constant. 

In  the  last  equation  consider  the  cases  when  e>l,  e<l, 
e  =  l. 

(3)  Show  that  C  +  y\cl>-\=iO  is  the  condition  that  (312) 
shall  represent  two  straight  lines,  real  or  imaginary. 

(4)  Find  in  how  many  ways  a  conic  passing  through  the 
four  common  points  of  two  given  conies  can  be  reduced  to  two 
right  lines,  using  the  condition  of  Exercise  3. 

(5)  Show  that,  if  two  conies  have  their  axes  parallel,  any 
conic  passing  through  the  common  points  of  these  two  will 
have  its  axes  parallel  to  theirs. 

(6)  Hence  show  that,  if  a  pair  of  right  lines  be  drawn 
through  the  four  common  points  of  a  circle  and  any  conic,  the 
bisectors  of  the  angles  between  these  two  lines  will  be  parallel 
to  the  axes  of  the  conic. 


Chap.  IV.]  SCALAR   POINT   EQUATIONS.  133 


CHAPTER   IV. 

SCALAR  POINT   EQUATIONS   OF   THE   SECOND   DEGREE   IN 
PLANE   SPACE. 

104.  We  need  only  consider  homogeneous  equations ;  for,  hj 
eq.  (224),  3j9,e  =  1,  so  that  a  term  of  any  degree  in  ^>  may  be 
raised  to  any  other  degree  by  multiplying  by  the  proper  power 
of  Sjje. 

Any  homogeneous  equation  of  the  second  degree  may  be 
written  in  the  form 

p!#  =  0, (325) 

in  which  <^  is  a  linear  self-conjugate  function ;  for  the  left-hand 
member  of  such  an  equation  can  always  be  reduced  to  the  sum 
of  such  terms  as  Aip\qi  -plqi  ;  that  is,  to  the  form 

^{Ap\q  'p\q')  =  \%lp\{Aq  •  p\q'  +  Aq'  .p\q)^ 

=  ^p\'^[A{q-2)\q'  +  q'-p\q)^=iy\<l>p, 
if  we  write 

<f>p  =  ^%lA(q-p:q'-\-q'-p\q)] (326) 

Of  course  we  may  have  q  =  q'  for  some  terms  of  the  summar 
tion. 

105.  Eq.  (325)  represents  a  curve  of  the  second  order ;  that 
is,  it  is  cut  in  two  points  by  any  right  line.  For,  let  p^xq^+yq^ 
be  the  equation  of  some  right  line,  and  substitute  in  (325)  : 

.-.   {xqy  +  yq.^\<i, (xq^  ■+■  yq^) 


whence        ^  =  "  gil'^ga  ±  V(gi|<^g2)'  -  9i|<^gi  •  g2l<^g2 

X  q2\<t>92 

^  -  qi<f><]2  ±  V-  qiq2\<f><liH2 (327) 

Q-^<t>Q2 


134  DIRECTIONAL  CALCULUS.  [Art.  106. 


As  y-  has  two  values,  it  appears  that  the  line  must  cut  the 

X 

curve  at  two  points. 

If  the  two  values  of  ^  are  equal,  the  line  must  be  tangent  to 


X 

the  curve.     The  condition  for  this  is 


qyq,\4>qM2  =  ^\ (328) 

ithi 
equation 


and  when  this  condition  is  satisfied,  -  = \-~^ :  so  that  the 

X  521^92 


^  gi  •  q^^i>q-  -  72  •  gi|<^g2  ..^29) 

qJ^<f>q2  -  qilHi  

gives  the  point  of  contact  of  the  line  with  the  curve  when 
(328)  is  satisfied. 

In  (328)  suppose  q.2  to  vary,  and  replace  it  by  p ;  then  the 
equation         ^^^j^,^^^^,  =  o   .    • (330) 

causes  p  to  be  always  on  a  straight  line  passing  through  the 
fixed  point  qi  and  tangent  to  the  locus  of  (325) .  As  (330)  is 
of  the  second  degree  in  p,  there  must  be  two  such  lines ;  i.e. 
two  tangents  to  the  locus  through  any  point  qi.  It  will  some- 
times be  more  convenient  to  write  the  equation  of  the  cutting 
line  in  the  form 

p  =  qi  +  yi; 

when  we  must  put  in  (327)  x  =  l,  and  c  for  q^,  thus  obtaining 
—  gii<^e  ±  V-  7if  0gi<^g /331X 

106.   Diameters.    To  find  the  locus  of  the  middle  points  of 
a  system  of  parallel  chords  of  the  locus  of  (325). 

In  (331)  let  qi  be  on  the  curve,  so  that  we  have  qi\<tiqi  =  0  ; 

then  y  =  ^^^ — ili^.     At  the  middle  point  of  a  chord  having 

the  direction  e,  we  have  j)  =  qi -{• -^  y^  =  qi rr^  •  «• 

.-.  j):«^c  =  5i|,/»c-gii</.£  =  0 (332) 


Chap.  IV.]  SCALAR   POINT   EQUATIONS.  136 

This  is  the  equation  of  a  diameter  conjugate  in  direction  to  c. 
Let  pi  and  p.,  be  any  two  points  in  this  diameter,  so  that  we 
have  pi\<f}€  =  0  and  p2\<f>€  =  0,  and  therefore  (^2— i>i)l^£  =  0. 
Now  P2  —Pi}  being  a  vector  along  the  diameter,  is  conjugate  to 
c;  hence  two  conjugate  directions  ci  and  t^  must  satisfy  the 
condition 

ci|<^c2  =  0 (333) 

107.    Tangent  and  polar.     Differentiating  (325),  we  have 

dp\<f>p  =  0. 

Hence  \<}>p  is  parallel  to  the  tangent  at  p ;  but  (335)  shows 
that  the  line  \<{>2)  passes  through  p;  consequently  \<f>p  is  the  tan- 
gent line  to  the  locus  at  p.  The  equation  of  the  tangent  line 
is  therefore 

^|#  =  0, (334) 

q  being  a  variable  point,  andjj  a  point  on  the  curve. 

If  e  be  some  point  not  on  the  curve,  let  us  determine  what 
line  \<f>e  is.  Suppose  tangents  to  be  drawn  from  e  to  the  curve, 
touching  it  at  pi  and  pj-  Then  these  tangents  will  be  \<f>pi  and 
\<j>P2.     But  as  they  pass  through  e,  we  must  have 

e\(f>2h  =  0=2h\<lie 
and  e|<^P2  =  0  =p2|<^e. 

These  conditions  show  that  |^e  passes  through  pi  and  p^,  the 
points  of  contact  of  the  tangents  drawn  to  the  curve  from  e. 
|0e  is  therefore  the  polar  of  e.  Let  q  be  any  point  on  |^e; 
then  we  must  have  q\<f)e  =  0  =  e\<f}q ;  so  that,  wherever  q  be 
situated  on  the  polar  of  e,  its  polar  always  passes  through  e. 

Thus  if  a  point  move  along  a  straight  line,  its  polar  passes 
through  a  fixed  point,  the  pole  of  this  line ;  and,  reciprocally, 
if  a  revolving  line  pass  through  a  fixed  point,  its  pole  moves 
along  a  fixed  line,  the  polar  of  this  point. 

Equation  (332)  shows  that  a  diameter  is  the  polar  of  a  point 
at  00.  Hence  the  polars  of  all  points  on  a  diameter  have  a 
common  point  at  00 ;  i.e.  they  are  parallel  to  the  diameter  con- 
jugate to  this. 


136  DIRECTIONAL  CALCULUS.  [Art.  108. 

108.  Center  of  the  locus.  The  center  is  at  the  intersection 
of  any  two  diameters ;  hence 

q,  =  m\(f>€i<f>€. (335) 

is  the  center,  ?>i  being  a  scalar  factor  so  taken  as  to  make  q^  a 
unit  point.  To  evaluate  m,  multiply  both  sides  of  the  equation 
into  3|e ;  therefore 

3  5e|e  =  1  =  3  m\(f>€i(f>€^  =  3  me<t>ei<f>e2 

so  that  (335)  becomes 

9e  =  J^r^ (336) 

If  we  have      ei^ei^cj  =  0, .     (337) 

while  |<^ci</>c2  is  not  zero,  then  the  center  of  the  locus  is  at  oc. 

If  q^  be  substituted  for  qi  in  (330),  we  have  the  equation 
of  the  tangents  passing  through  the  center,  that  is,  of  the 
asymptotes,  viz. : 

m|<^i></»9e  =  0 (338) 

109.  Conjugate  points.  k.nj  set  of  three  points  which  fulfil 
the  conditions 

qi\<f>q2  =  q2\H3  =  Qs\Hi  =  ^ (339) 

is  a  set  of  conjugate  points.  These  equations  cause  each  point 
to  be  on  the  polar  of  each  of  the  others ;  that  is,  the  points  are 
the  vertices  of  a  self-conjugate  triangle,  in  which  each  side  is 
the  polar  of  the  opposite  vertex  with  reference  to  the  curve 
p\<i>p  =  0. 

There  is  an  infinite  number  of  such  sets  of  points ;  for  take 
any  point  in  the  plane  of  the  curve  as  q^,  then  any  point  in  the 
polar  of  qi  as  q-2,  whose  polar  will  pass  through  q^,  by  Art.  107, 
and  will  cut  the  polar  of  q^  in  g,,.  If  one  point,  say  q^,  is  at  oc, 
the  other  two  will  be  on  a  diameter ;  if  q.i,  be  also  at  co,  then  ^i 
will  be  at  the  intersection  of  two  diameters,  that  is,  it  will  be 
at  the  center;  thus  q^,  cj,  ca  form  a  conjugate  system  if  we  have 
9c|^«i  =  «i|<^«2  =  ^^.^Qa  ^-nd  we  see  that  conjugate  directions  are 
only  a  particular  case  of  conjugate  points. 


Chap.  IV.]  SCALAE    POINT   EQUATIONS.  137 

110.  Normal  system  of  conjugate  2)oints.  If  a  system  of 
conjugate  points,  besides  the  conditions  (339),  satisfy  also  the 
conditions 

QMa  =  q2\qs  =  Qs\Qi  =  0, (340) 

they  may  be  called  a  normal  system.  We  proceed  to  show, 
that  with  reference  to  any  curve  represented  by  the  equation 
p'<f>2)  =  0,  there  is  one,  and  only  one,  normal  system  of  conju- 
gate points. 

111.  Solution  of  the  equation  p<f>p  =  0.  This  equation  is 
equivalent  to  <f>2^  =  np,  or  (^  —  n)j)  =  0.  Multiply  the  com- 
plement of  the  first  member  by  any  three  points  g„  q^,  q^,  and 
we  have  three  scalar  equations  equivalent  to  the  single  non- 
scalar  equation,  viz. : 

'qiX<f>-n)p=2y\(<l>-n)qii=0' 

q2i{i>-n)p=2}{<f,-n)q.  =  0  I (341) 

Qs[(i>  -  ^i)P  =P\ (<i>  -  n)qs  =  0 
Each  of  these  equations  must  be  satisfied  by  the  same  values 
of  2^  which  satisfy  the  given  equation ;  i.e.  they  are  simulta- 
neous equations.  The  point  p  must  therefore  be  simultaneously 
in  each  of  the  three  lines  \{4>  —  n)qiy  \(<t>  —  n)q2,  |(^  — w)g3, 
which  requires  that  these  lines  shall  have  a  common  point,  the 
condition  for  which  is 


{<)>  -  n)qi{<i>  -  n)q2(<l>  -  n)qs  =  0 
or  n^  —  fcgn^  +  kin  —  A-q  =  0 


[,....     (342) 
(343) 


in  which  Av,  =  (fiqi4>Q-2'^Q3  -^  Qi<1^3 

h  =  {qiH2<i>q3+q2<l>q3<l>qi+g!i<l>QiH2)  -^qiq^z 
h  =  {qiq^Ha+q^My+q^qMi)  -^  Q'iMs 

The  fc's  are  invariants;  i.e.  they  have  the  same  values  what- 
ever position  the  points  ^i,  gg*  Q's  n^^-J  occupy,  which  may  be 
shown  as  in  Art.  98  in  the  case  of  w-o  and  wii.  The  solution  of 
(342)  will  give  three  values  of  n,  which,  substituted  in  (341), 
will  give  the  required  points  at  which  ^p  =  np.     Let  the  roots 


138  DIRECTIONAL  CALCULTTS.  [Art.  112. 

of  (342)  be  Wj,  n^,  n^,  and  let  the  corresponding  values  of  p  be 
Pi)  P2,  Ih ;  then,  by  (341),  the  equations 

pM  -  ni)qi{<i>  -  n{)q2  =  0  "j 

2h\{<l>-n,)gr(cf>-n,)q,  =  0  V    ....     (344) 

Ps\ (<f>  —  '>h)qii<l>  -  ^3)5^2  =  0  J 
give  the  points  pi,  p^  2h- 

112.  To  show  that  the  points  just  determined  form  a  nor- 
mal conjugate  system.  We  must  have  «^pi  =  Uij^i,  <l>p-2  =  ^dhy 
<f>p^  =  W3P3.  Now,  as  the  function  (f>  is  self-conjugate,  wi'ite  it 
in  the  most  general  form  of  such  a  function  in  terms  of  the 
p's ;  that  is, 

2«/>p  =  Wi(pi  'P2P3P  4-|P2i>3  'Pi\p)  +  ihiPi  -PsPiP  -hhhPi  -P2\P) 

+  ^3(^3  -PiP-P  -h\Pi2^2  -PsIp)- 

If  this  value  of  <f>  satisfies  the  conditions  above,  we  must 

have  P1IP2  =  P2IP3  =  PalPi  =  0  and  P1P2P3  =  Pr  =  2^  =  2h-  =  1  • 
These  conditions  give  at  once 

2  <f>pi  =  Wi(pi  -i-lPiPa),  etc. ; 
but  they  also  cause  pi  to  be  on  the  lines  \p2  and  \ps  simulta- 
neously, so  that  pi  =  m\p22h)  '''-  being  some  scalar  constant. 
Hence  Pi\pi  =  mpi2hP3i  or  m  =  1,  so  that  the  required  condi- 
tion ^pi  =  niPi  is  satisfied ;  and  so  for  the  others. 

Finally  p^^P2  =  n2Pi\2h  —  ^,  etc.,  so  that  all  the  conditions 
of  (339)  and  (340)  are  satisfied,  and  hence  the  points  2hi  2>2)  2h 
form  a  normal  conjugate  system.  It  appears  then  that,  what- 
ever be  the  original  form  of  ^,  one  set  of  three  points  may 
always  be  found  such  that,  if  these  be  taken  for  reference 
points,  ff>  is  reduced  to  the  form 

<f>p  =  n^2h-2^\2h  +  n2P2-I)\P-2  +  'nzPi'P\P3-  •  (345) 
Note  that  these  will  not  in  general  be  all  unit  points ;  for  if 
we  express  pi,  P2,  Ps  in  terms  of  the  original  reference  points, 
we  have  nine  constants,  and  have  subjected  the  points  to  seven 
conditions,  so  that  we  cannot  apply  the  three  additional  condi- 
tions necessary  for  unit  points. 


Chap.  IV.]  SCALAR    POINT   EQUATIONS.  139 

The  three  j)oints  above  determined  will  always  be  real,  i.e. 
the  roots  of  (342)  are  always  real;  for,  suppose  one  to  be  im- 
aginary, and  call  it  n  +  n'i,  and  the  corresponding  value  of  p, 
p  +p'i,  in  which  i  —  V— 1 ;  then 

^{P+ip')^{n  +  in'){p  +  ip')- 
or,  equating  separately  to  zero  real  and  imaginary  parts, 

(^p  =  np  —  n'j)',    4>P'  =  ''^'P  +  ^U^'- 

•'•  P'l^P  =  wj>'|i^  —  n'p)'-  =  p\<l>p'  =  n'jT^  +  np\p'. 

...  n'{p^+p'-^)  =  0, 

which  can  only  be  satisfied  by  n'  =  0,  so  that  there  can  be  no 
imaginary  value  of  n  or  ji. 

113.  Canonical  form  of  p\<f>p.  With  the  form  of  <f>  given  in 
(345)  we  have 

p\<f>p  =  n,(p\p,y  +  n,(p\p,)-  +  ns(p\ps)%      .     (346) 

which  is  the  canonical  form  of  the  scalar  quadratic  in  p,  and 
we  have  shown  that,  whatever  may  have  been  the  original 
form  of  |)|<^j),  it  can  always  be  reduced  to  this  canonical  form 
by  properly  choosing  the  reference  points.  The  first  two  terms 
of  (346)  may  be  written 

i>|(i^iV?ii  +P2V-  n^)  -pKih^ni  -iW-  W2), 

or  23\Qi '  p\Q2j  if  we  put 

qi=2h'^</^i+2h^—'>h  and  q.,  =  Pi^n^  —p-f^  —  n^. 

Thus  the-  equation  of  the  locus  becomes 

p\qx'P\q2  +  n,{p\p,y==() (347) 

In  this  form  the  eqiiation  shows  that  the  curve  is  tangent 
to  |gi  and  \q.2  at  the  points  where  they  are  cut  by  \pi.  But,  by 
the  last  article,  2h\qi  =  ^5  and  ps\q.,  =  0,  so  that  \qi  and  \q2  pass 
through  P3,  and  touch  the  curve  where  it  is  cut  byjpiP2=  \P3' 


140  DIRECTIONAL   CALCULUS.  [Art.  114. 

114.  Condition  that  p\<f>p  shall  break  up  into  two  factors  of  the 
first  degree.    If  this  be  possible,  it  will  take  the  form 

p\<i>p=p\qi-p\q^ 

whence  <^P  =  i (gi  •  p\q2  +  (h  •  p\qi)- 

If  we  take  any  three  values  of  p,  as  p)\i  Jhf  Ihi  the  points 
#>ij  ^Ih,  <\*lh  will  all  be  on  the  line  q^q.2,  so  that  their  product 
will  be  zero.     Conversely,  if  we  have 

<t>Pi<t>P2'j>Ps  =  0 (348) 

for  any  three  points  whatever,  jh^  Ihj  Ihi  then  the  function  <^ 
must  have  the  above  form,  and  p\<i>i^  is  factorable.  The  ex- 
pression <t>Pi<l>2hi>2h  is  the  discriminant  of  p\^p,  and  is  the  same 
as  P1P2P3 '  ko  of  eq.  (343). 

115.  Nature  of  the  locus  at  00.  To  deterinine  this  we  will 
find  the  intersection  of  the  line  at  00  with  the  locus.  In  (327) 
put  ci  and  Co  for  q^  and  q^,  thus  obtaining 


y_  —  ei|^£2  ±  V—  eie2!<;^ei<^e2 

Now,  as  £1  and  cg  n^ay  be  any  points  at  00  whatever,  write 
ci  =  gj  —  ^0,  £2  =  62  —  «o ;  then 

£l€2  =  6163  +  ^2^0  +  CoCl  =  I  (?0  +  ^1  +  e,)  =  3|g, 

so  that  the  quantity  under   the  radical  becomes  —  3  et^t^^to. 
Thus  the  two  vectors 


£2[^£2  •£!  +  (—  C]|^C2  +  V—  3  e</>£i^£2)  €2 


and  £2l<^c2-«i  — (eii<^e2+V— 3e</>£i</)£2)£j  .     .     (349) 

are  respectively  in  the  direction  of  the  two  points  at  oo  of  the 
locus.  These  vectors  are  imaginary,  parallel,  or  real,  accord- 
ing as  e<^£i^£2  is  -f ,  0,  or  — ,  corresponding  respectively  to  no 
real  points  at  go,  two  coincident  points  at  oo,  and  two  real  points 
at  X.     In  the  second  case  the  vectors  of  (349)  are  parallel  to 


Chap.  IV.]  SCALAR   POINT   EQUATIONS.  141 

the  axis  of  the  parabola ;  in  the  third  case  they  are  parallel  to 
the  asymptotes  of  the  hyperbola.  In  the  first  case  the  curve 
is  an  ellipse.     We  have  thus 

positive  for  an  ellipse 


e<^ei<^€2 


zero  for  a  parabola 
negative  for  an  hyperbola 


(350) 


116.  The  most  general  form  of  the  homogeneous  second- 
degree  equation  in  p  may  evidently  be  written 

A{p\e,y'  +  B{p\e,y  +  C{p\e,Y  +  2A'p\e,  -pie, 

+  2B'i>|e2.i)ieo  +  2C"i)|eo-p|ei  =  0    .     (351) 

\i  all  the  points  involved  except  the  variable  point  p  are  ex- 
pressed in  terms  of  the  reference  points ;  for  no  other  combi- 
nations can  be  made  of  the  three  quantities  p\eQ,  p\ei,  p\e2,  which 
shall  be  of  the  second  degree.  As  there  are  five  arbitrary 
constants,  the  curve  may  be  subjected  to  five  arbitrary  condi- 
tions.    Write 

<^i)  =  {Ae^  4-  C'ei  +  B'e2)p\e^  -f  (  C'e^  -f  Be^  +  A'e,)p\e, 

+  iB'e,  +  A'e,  +  Ce,)p\e„      (352) 

and  (351)  becomes  p\(f>2)  =  0. 
If  we  write 

Slco'  =  Aeo  +  C'e,  +  B'e,  ^ 

SBe/  =  Ceo  +  Be,  +  A'e,  L (353) 

Qe,'  =B'eo-\-A'e,  +  Ce,  J 

we  have  (f>p  =  ^e^)' ■  p\6q -\- ^ei  '  p\ei -\- Qe^' '  ple^,  .  .  (354) 
and  (351)  becomes 

^Ph'  •  i>|eo  +  ^pW  •  p\e,  +  ©Plea'  •  ple^  =  0.    .     (355) 

117.  Curve  through  the  reference  points.    Let  A  =  B=:C=0 

in  (351)  ;  omit  the  primes  and  divide  by  2,  and  the  equation 
becomes 

Ap\ei .  p|e2  +  Bp\e.^ .  p\eo  +  Cp\e«  •  pie,  =  0.       .     (356) 


142 


DIRECTIONAL  CALCULUS. 


[Art.  118. 


As  this  equation  is  satisfied  when  p  is  replaced  by  Pq,  Cj,  or 
fij,  it  follows  that  the  curve  passes  through  the  reference  points. 
It  is  the  most  general  form  of  the  equation  of  a  conic  through 
these  points,  because  no  other  term  can  be  formed  which  will 
vanish  under  each  of  these  conditioiis.  It  is  the  same  equation 
as  (246)  if  Li,  Zj,  L^  be  taken  as  sides  of  the  reference  tri- 
angle. 

The  equations  of  the  tangent  lines  to  (356)  at  the  reference 
points  are  p[<^eo  =  0,  etc. ;  and,  by  (353), 

^e^  =  Cci  +  Be.2,  t^e^  =  Ae.,  +  C^o,  <^e^  =  Be^  +  Aei. 

Hence  these  tangent  lines  may  be  written 

j>|ei  ,  ^2  _  P^  ,P\^  _ V\^  ,  i>  ei  ^0  (-357^ 

B^  C  ~  C  '^  A       A       B         '  "^       ^ 


118.  Circle  through  the  reference  points.  Let  the  circle  be 
drawn  through  the  reference  points  as 
in  the  figure,  and  also  the  tangents  at 
these  points.  Let  oq,  aj,  05  be  the  an- 
gles of  the  triangle  as  shown,  and  Oo, 
ai,  02  be  equal  respectively  to  T{eie^, 
T{e^(^,  and  T(eQei).  Let  q  be  any 
point  of  the  tangent  at  631  ^-^^d  let  go» 
qi,  be  the  feet  of  the  perpendiculars 
from  Co,  ei  on  the  tangent  at  e,.  Then 
we  have 

t^=  _£ko_  y(ei9i)_aosinao^ao^ 
^0^2^  ~~      q\ei      T{e,^o)      «i  sin  oj      a/ 

Hence  the  equation  of  the  tangent  at 

62  is  ii^  -(-  ?Li  =  0.     And  by  symmetry  the  tangents  at  e^  and 


a„ 


a,- 


Ci  are 


a"  "1         2  —  — V  "r  — n  —  ^. 

ai'      a^       ui       «o 


Chap.  IV.] 


SCALAR   POINT   EQUATIONS. 


143 


Comparing  these  equations  with  (357),  it  appears  that,  if 
(356)  is  to  represent  a  circle,  we  must  have 


A 


a/ 


(358) 


119,  Exercises.  —  (1)  Verify  the  result  of  the  last  article  by 
transforming  (356),  subject  to  the  condition  (358),  to  a  vector 
system  by  Art.  75. 

(2)  Show  in  the  same  way  that  the  equation 

(ai'  +  a/  -  flo-)  {P\e,y  +  {(h'  +  ««'  -  a,')  {p^Y 

+  (ao^  +  «r'-a/)(i^;e,)'  =  0 
represents  a  circle,  which  is  real  or  imaginary  according  as  the 
reference  triangle  has  an  obtuse  angle,  or  has  all  its  angles 
acute. 

(3)  Show  that  the  equation  pc|<^p^c  =  0  represents  the  two 
tangents  to  the  conic  parallel  to  c. 

(4)  Hence  find  the  tangents  to  (356)  parallel  to  the  sides  of 
the  reference  triangle. 


Ans.  Tangents  ||  to  Cffii  are  p|(eo  +  e^)  =  ±p\e. 


H^%- 


(5)  If  two  conies  through  the  reference  points  have  respec- 
tively the  coefficients  Ai,  B^  Ci  and  A^,  B.^,  C.^,  show  that  their 
fourth  common  point  lies  on  the  lines 


Bq    Co 


P\eo' 


Cg  A2 


p\ei  = 


Ao  bJ^I^ 


(6)  Apply  (336)  to  finding  the  center  of  the  curves  repre- 
sented by  (351)  when  A'=B'=C'=0,  and  when  A=B=C=0. 

.        BCeo±CM±ABe,, 
"*''•       BC+CA  +  AB     ' 

A'{B'-\-C'-A')eo+B'(C'+A'-B')e,  +  C'(A'-\-B'-C').e^ 
4:A'B'-(A'-i-B'-C'y 


144  DIRECTIONAL  CALCULUS.  [Art.  120. 

(7)  Determine  the  nature  of  the  curve  represented  by  (351) 
under  the  following  conditions  : 

(a)  A  =  B  =  C=-.0,  A'  =  B'  =  C'  =  1. 

(b)  A  =  B  =  C  =  0,  A'  =  B'=  -C'  =  l. 

(c)  A  =  B  =  C=0,  A'  =  4,  B'  =  C'  =  1. 

(d)  A  =  B  =  C=0,  ^'  =  3,  5' =  4,  C'  =  o. 

(e)  A  =  B=-C=1,  A'  =  B'=C'  =  0. 
{f)A=-l,  B=2,  C=S,  A'  =  B'  =  C'  =  0. 
(g)  A  =  B  =  C=  -1,  A'  =  B'  =  C'  =  1. 

In  each  of  the  above  cases  find  the  center,  directions  of  the 
asymptotes  when  real,  and  the  points  in  which  the  sides  of 
the  reference  triangle  cut  the  curve. 

(8)  Discuss  the  equations 

and  pCf^LiexL^^P  =  ^^ 

(9)  If  a  triangle  be  inscribed  in  a  conic,  the  tangents  at  the 
vertices  cut  the  opposite  sides  in  three  coUinear  points. 

(10)  If  a  triangle  circumscribe  a  conic,  the  lines  joining  the 
vertices  with  the  points  of  contact  of  the  opposite  sides  have 
a  common  point. 

(11)  If  a  line  be  tangent  to  a  conic  whose  equation  is 
p\<t>2P  =  0,  find  the  locus  of  its  pole  with  reference  to  any 
other  p\<l>iP  =  0. 

Ans.  q\<f>2~^<l>i'Q  =  ^- 

120.  TJie  equations  p\<l>p  =  C  and  p\(f>p  =  0  represent  conies 
zvhich  are  concentric,  similar,  and  similarly  placed.  For,  since 
3p|e  =  l,  we  may  write  the  first  equation  p\<f>p  =  C(Sp\ey 
without  changing  its  meaning ;  but  p\e  =  0  is  the  equation  of 


Chap.  IV.]  SCALAR  POINT  EQUATIONS.  146 

the  line  at  x ;  hence  the  equation  in  its  present  form  is  that 
of  a  conic  tangent  to  j)\i*P  =  0  at  the  points  where  it  cuts  the 
line  at  oc ;  that  is,  the  two  curves  have  the  same  asymptotes, 
real  or  imaginary,  which  proves  the  proposition.  It  will  also 
appear  on  examination  that  the  expression  for  the  center  will 
be  the  same  for  the  two  curves,  if  we  note  that  cje  is  always 
zero,  being  the  product  of  a  point  at  oo  into  the  line  at  oo  . 

121.  Anti-polar  of  any  point.  Let  e  be  the  point  whose  anti- 
polar  is  to  be  found,  and  let  e'  be  so  situated  that  q^  =  ^(e+e')  ; 
i.e.  the  center  of  the  curve  is  midway  between  e  and  e'  on  the 
line  joining  them.  Then  the  polar  of  e'  will  be  the  anti-jjolar 
of  e ;  and  since,  by  the  given  relation,  e'  =  2  q^  —  e,  we  have  for 
the  required  equation 

p\<f>(2q,-e)  =  0 (359) 

122.  Reciprocating  ellipse.  We  proceed  now  to  find  the 
equation  of  the  ellipse  referred  to  in  Arts.  41-4A,  with  refer- 
ence to  which  each  reference  point  is  the  anti-pole  of  the 
opposite  reference  line. 

When  the  reference  triangle  is  equilateral,  the  reciprocal 
ting  curve  is  a  circle.  Now  the  equation  of  a  circle  through 
the  reference  points  is,  in  this  case, 

pVi  •i>le2  +pje2  -pK  +^1^0  •P\ei  =  0, 

because  aif  =  ai  =  a*  Hence,  by  Art.  120,  the  reciprocating 
circle  will  be 

p\ei-p\ei+p\e2-p\eo-\-p\eo-p\ei  =  C{3p\ey    .     (360) 

if  C  be  properly  determined.  We  may  infer  that  the  equation 
of  the  reciprocating  ellipse  will  have  the  same  form  when  e^^e^ 
is  not  an  equilateral  triangle. 

When  A  =  B  =  C=0  and  A' =  B' =  C  in  (351),  we  have 
q^  =  e,  so  that  (359)  becomes  p\<f>{2e  —  e)  =  0. 


146  DIRECTIONAL   CALCULUS.  [Art.  123. 

From  (360)  we  have 

#  =  i(ei  +  62)  -P^  +  ^(62  +  ^0)  •i>!ei 

+  |-(eo  +  ei)  ■pe.  —  ^Ce-ple, 

whence    0eo  =  i(ei  +  e-i)  —  3Ce,  etc. 

Now  if  any  side  of  the  reference  triangle,  as  e^Co,  is  to  be  the 
anti-polar  of  the  opposite  vertex  e^,  we  mvist  have 

eie2\<f>{2e  —  Co)  =  0,  or  |eo</)(2e  —  e„)  =  0. 

That  is, 

\eo(f> (2 ej  +  2 e.y  —  Cy)  =  jeoCe.  +  ^o  —  OCe  -f  Co  +  ei  -  6Ce 

-^(ei  +  ^.)  +  3(7e) 

=  i(^.-ei)- 3(7(62 -e,)  =  0. 

Whence  0=^. 

It  is  evident  from  the  symmetry  of  the  equations  that  the 
equations  \ei<f>{2e  —  Cj)  =  0  and  \e2<i>(2e  —  e,)  =  0  will  give  the 
same  value  of  C;  hence  the  equation  of  the  reciprocating 
ellipse  is 

6{p\ei'p\es+p\e2-p\€o+p\eo-p\ei)-{32)\e)-  =  0.     .     (361) 

123.  Complement  of  any  point.  We  will  give  now  the  proof, 
referred  to  in  Art.  44,  that  \p  is  the  anti-polar  of  p  with  refer- 
ence to  (361).    Let 

p  =  Zcq  -|-  wifii  +  ne2 ; 
then  its  anti-polar  is 

\<t>{2e  -  leo  —  mei  —  ncj)  =  |<^[(f  -  0«o  +  (|-  wi)ej  +  (|  -  71)62], 

0  being  derived  from  (361).  We  find  </>6o  =  3(ei-f-e2  — e), 
^gj  =  3(62  -I-  Co  —  e),  «/»e2  =  3(eo  +  61  —  e).  Now,  if  the  proposi- 
tion is  true,  we  ought  to  have 

1  (ico  +  me,  +  ne.)  <^ [(2  -  3  Oco  +  (2  -  3  m)e,  +  (2  -  3  n)e.^  =  0. 
.♦.  |(Zeo  +  wei  +  7<62)[ (2-30  («i  +  ^2  -  e)  +  (2  -  3  m)(e.,  +  60-e) 

+  (2-3n)(eo-f6.-e-)] 
=  K^^o  +  tnei  +  7160)  (3  ko  +  3  me,  -f  3^62)  =  0. 
Hence  the  proposition  is  demonstrated. 


Chap.   IV.]  SCALAR   TOINT   EQUATIONS.  147 

124.  Line  equations.  If,  in  eq.  {ooQ),  we  write  L  for  \p, 
we  have 

ALei-Le.  +  BLe.,'Le(,+  CLeQ'Le^  =  Q,     .    .     (362) 

the  equation  of  a  conic  enveloped  by  L,  and  tangent  to  the 
sides  of  the  reference  triangle,  because  it  is  satisfied  when 
Jj  =  €162,  or  ^2^0?  01'  ^o^i- 
Write 

xl,L  =  I {Be^  +  Ce.)  •  Le^  +  j (Cco  +  Ae.^  •  Le^ 

+  [{Ae,-^  B€^)-Leo,    .     (363) 
and  (362)  becomes 

L\il/L  =  0 (364) 

Comparing  with  Arts.  116  and  117,  it  appears  that 

^L  =  \<f>i)  =  ]cf>  L (365) 

We  can  thus  pass  at  once  from  any  point  equation  to  its 
complementary  line  equation,  or  the  reverse. 

The  function  \jj  is  a,  linear  line  function  of  L,  self -con jugate, 
and  therefore  possessing  all  the  properties  shown  to  belong  to 
<f).     Diiferentiating  (364),  we  have 

dL\il/L  +  L\il/dL  =  2  dLl^pL  =  0. 

Hence,  by  Art.  79  and  eq.  (364),  \\)/L  is  a  point  on  L,  and 
also  on  dL,  a  line  through  e  and  the  point  of  contact  of  L  with 
the  curve.  Consequently  \if/L  must  be  itself  the  point  of  con- 
tact of  L. 

If  L  be  not  tangent  to  L\ij/L  =  0,  then  \iJ/L  is  the  pole  of  L. 

125.  Center  of  L\\l/L  =  0,ij/  being  any  linear  self -conjugate  line 
function.  The  center  of  any  conic  is  the  pole  of  the  line  at  00. 
Hence 

mq,  =  \ij/\e  =  4>e, 

m  being  a  scalar  constant.     Multiply  the  complement  of  this 
equation  by  3  e ;  and  we  have 

3  nie\qc  =  m  =  3e\<f>e. 


148  DIRECTIONAL   CALCULUS.  [Art.  126. 

.-.  5„  =  <^e-=-3ej</)e  =  ji/'C-T- 3ei//]e (366) 

If  we  have 

e|«/)e  =  0,     .    .     .    , (367) 

the  center  is  at  co,  and  the  curve  is  a  parabola.  The  condition 
(367)  makes  the  curve  2)\<f>jy  =  0  pass  through  e,  the  center  of 
the  reciprocating  ellipse.  Thus,  when  a  curve  passes  through 
the  mean  point  of  the  reference  triangle,  its  reciprocal  curve 
is  a  parabola. 

126.  Determination  of  the  cicrve  L\\pL  =  0.  If  real  tangents 
can  be  drawn  to  the  curve  jjj^p  =  0  through  e,  then  the  recipro- 
cal curve  (364)  must  have  two  real  points  at  go,  viz. :  the  anti- 
poles of  these  two  tangents,  and  must  therefore  be  a  hyperbola. 
If  no  real  tangents  can  be  drawn  to  p\^p  =  0  through  e,  then 
(364)  has  no  real  points  at  go,  and  is  therefore  an  ellipse.  If 
two  coincident  tangents  can  be  drawn,  the  curve  is  a  parabola, 
as  was  shown  in  the  last  article.  Xow  (330)  was  shown  to  be 
the  equation  of  the  tangent  lines  to  p\<^p  =  0  through  q^ ; 
hence,  putting  e  for  g„  the  tangents  through  the  mean  point  are 

pe\(f>p<}}e  =  0  =  e|<^e  •p\<f>p  —  (p!<^e)^ ; 
or,  writing  ^ij)  =  fpj)  •  e\cf>e  —  (fte  •  p\<{>e,       ....     (368) 

the  equation  becomes 

p\<l>iP  =  0 (369) 

The  two  lines  represented  by  (369)  will  be  real,  coincident,  or 
imaginary,  according  as  they  cut  the  line  at  co  in  real  coin- 
cident or  imaginary  points ;  that  is,  by  (350),  according  as 
e<f>i€i<f>i€2  is  negative,  zero,  or  positive.  Hence  we  have  the 
criterion  for  eq.  (364), 


e<^jCx«/>i€2 


-f-  for  an  ellipse 
0  for  a  parabola 
—  for  a  hyperbola 


(370) 


Chap.  IV.] 


SCALAR   POINT   EQUATIONS. 


149 


Let  Ci,  •••  e^,  be  any  five  fixed  points ; 
P 


127.  PascaPs  theorem. 
to  find  the  locus  of  p 
when  q^,  q.y,  q^  are  in 
one  straight  line,  as 
shown  in  tlie  figure. 
The  condition  for  this 

and  we  have 

q^  =  e^'  e^p. 
Hence  {pe^  -  e^^)  {e^e^  •  6465)  (6263  •  er,p)  =  0. 


(371) 


This  is  a  scalar  equation  of  the  second  degree  in  p.  It  there- 
fore represents  a  conic.  It  is  evidently  satisfied  when  p  =  ei, 
and  when  p  =  e^.  Let  p  =  e2',  then  g^  and  ^2  ^-^e  each  on  the 
line  6162,  and  q^  is  the  point  €2 ;  hence  the  equation  is  satisfied. 
Let  p  =  63 ;  then  qi  and  q^  each  coincides  with  e^,  so  that 
9i5'25'3  =  0.  Finally,  letj:>  =  e4;  then  q^  coincides  with  e^,  and 
q^  and  q^  are  both  on  the  line  6465,  so  that  qiq2qs  =  0  in  this  case 
also.  Thus  the  conic  passes  through  the  five  fixed  points,  and 
the  hexagon  j),  ^u  •  •  •  e^  is  inscribed  in  a  conic.  We  have,  there- 
fore, the  following  theorem : 

If  a  hexagon  be  inscribed  in  a  conic,  the  pairs  of  opposite  sides 
intersect  each  other  in  three  collinear  points. 

128.  Brianchon's  theorem.  This  theorem  is  the  complemen- 
tary, or  reciprocal,  of  Pascal's,  and  will  be  obtained  by  writing 
lines  for  points  in  (871).     It  may  be  stated  as  follows : 

If  a  hexagon  be  circumscribed  about  a  conic,  the  three  lines 
joining  the  opposite  pairs  of  vertices  will  pass  through  a  common 
point. 

129.  Inversion  of  (f>.  Let  ^p  =  e,  so  that  we  have  also 
J)  =  <t>~^e,  and  let  ^^  be  the  function  conjugate  to  <f>,  so  that 


150  DIRECTIONAL  CALCULUS.  [Akt.  129. 

q\<l3p  =  2^\<t>cQ-     Also,  let  qi,  q^,  q^  be   any   three  non-collinear 
points.     Then 

qi\<f>P  =p1Mi  =  ?ile>    <12\4>P  =lMcq-z  =  q2\e,    qs\4>P  =p\<f>cq3  =  ^sle- 

In  eq.  (102)  put  </><.^i  for  2h,  <^cQ'2  foi^  Pd  and  ^.g,  for  p^ ;  then, 
noting  the  equations  just  given,  we  have 

^-h=p=- — ,     ,    [|<^cg2</>cg3  •  ^i|e  +  \4>cqs^cqi  ■  q2\e 

<l>.qri>.q2^cqs  -^  \^.qr<t>.q2  -  q^lel     (372) 

When  </>  is  seZ/-con jugate,  of  course  <^c  =  <^-  If  e  =  jga^a) 
(372)  becomes 

<A~%2^3-Mlte</>c?3  =  fyi?2g3-|<^c92M3 (373) 

As  the  sum  of  any  two  linear  functions  is  itself  a  linear 
function,  and  its  conjugate  is  the  sum  of  the  conjugates  of  the 
two  functions,  we  can  invert  such  a  function  by  (372) .  Take 
for  instance  (^  +  w)j9,  of  which  the  conjugate  is  (^<.  +  n)p, 

and  we  have 

\{<l>c+n)q2{cl>,+n)qs . p\qi  +  \(<l>,+n)q^(<j>,+n)qi  •p\q2 

.,   ,  „x-l.,     +i(«^c  +  H)gi(<^,+H)g2-ij|g3 

^'^'^   ^    ^  iqc  +  n)q,{<l.,+n)q,{cf>,+7i)q, 

-  __  K<i>~^p  +  t^xP  +  '^iP 

ko  +  Jcin  +  k2n^  +  n^' 
in  which  Jcq,  ki,  k^  have  the  values  given  in  (343)  with  <^„  sub- 
stituted for  ^,  and  xP  is  a  linear  function  of  p,  the  coefficient 
of  n  in  the  expansion  of  the  numerator  of  the  second  member 
of  the  equation. 

Clearing  of  fractions  and  operating  by  ^  +  n,  we  have 

{ko+\n+k2n^+n^)p=kf,p-itn{^XP+^o'i>~^P)+^\'^-^X)P+'^^h'>- 
This  equation  must  be  true  for  all  values  of  n,  and  therefore 
the  coefficients  of  different  powers  of  n  must  vanish ;  hence 

from  which 

XP  =  (^'2  -  ^)p, 
and  koi>-^p  =  {ki-k.2cfi-{-cf>^)p (374) 


Chap.  IV.]  SCALAR   POINT   EQUATIONS.  161 

Thus  we  have  another  inversion  formula.  If  we  substitute 
the  value  of  x  j^st  found  in  that  of  (^  +  n)~^  as  written  above, 
we  have  a  formula  sometimes  useful,  viz. : 

Finally,  operating  by  fj>,  (374)  may  be  written 

•  {<l>^-Jc,^'  +  k,i>-Jco)2^  =  0 (376) 

130.  Exercises.  —  (1)  Invert  <^p  as  given  in  eq.  (352)  with 
A  =  B=C=0.    Also  with  A'  =  B'  =  C'  =  0. 

(2)  Invert  <^J3=  mp  +  ej  -plea 

Ans.  t^-'^p  =  —  [mci  •  j^Iei  +  (eg  —  e{)  •  ple^  +  e^m  ■  p\eo']. 

(3)  Show  that  if  the  sides  of  a  triangle  pass  through  three 
fixed  points,  and  two  of  the  vertices  move  on  fixed  right  lines, 
then  the  third  vertex  describes  a  conic. 

Ans.  Equation  of  conic  is  ppiLip^L^p^p  =  0. 

(4)  Show  that  if  the  vertices  of  a  triangle  move  on  three 
fixed  right  lines,  and  two  of  the  sides  pass  through  fixed 
points,  then  the  third  side  envelops  a  conic. 

Ans.  Line  equation  of  conic  is  LLiPiL^p^L^L  =  0. 

Pascal's  and  Brianchon's  theorems  can  be  derived  from  (3) 
and  (4)  respectively. 

(5)  When  the  three  points  of  Ex.  (3)  are  collinear,  show 
that  the  locus  reduces  to  two  straight  lines. 

(6)  When  the  three  lines  of  Ex.  (4)  have  a  common  point, 
show  that  the  envelope  reduces  to  two  points  and  the  straight 
line  joining  them. 

(7)  Write  the  equation  of  a  conic  passing  through  four 
points  and  tangent  at  one  of  them  to  a  given  right  line. 

Ans.   (pei  ■  6364)  (6162  •  6464)  (6363 .  e^p)  =  0. 


152  DIRECTIONAL  CALCULUS.  [Art.  130. 

(8)  Write  the  equation  of  a  conic  through  three  points  and 
tangent  at  two  of  them  to  given  lines. 

Ans.   (pei  •  60^3)  (^ici  •  egts)  (6263  •  e4p)  =  0. 

(9)  Write  the  line  equation  of  a  conic  tangent  to  four  right 
lines,  the  point  of  contact  being  given  on  one  of  them. 

Ans.  Lines  are  Li,  Lo,  Lg,  6464,  and  point  64,  and  equation  is 

[LL, .  (L3 .  64^4)]  [L,L, .  64]  [L,L, .  (64^4  •  i)  ]  =  0. 

(10)  Write  the  equation  of  a  conic  tangent  to  three  given 
lines,  the  points  of  contact  being  given  on  two  of  them. 

A71S.  If  lines  are  efy,  Lo,  CgCg,  and  points  gj  and  63,  then  the 
equation  is  [(iyei£i)(i,e3f3)]eie3[(eici  •  L.^  (egCg  •  i)]  =  0. 

(11)  If  L,  x>,  <f>,  and  ip  are  related  as  in  Art.  124,  show  that 
L\\l/L  =  0  and  p\<li~^p  =  0  are  the  line  and  point  equations  of 
the  same  curve,  viz. :  the  anti-polar  reciprocal  of  p\4>2i  =  0. 
Also  that  L\\p~'^L  =  0  and  J3|<^p  =  0  are  the  line  and  point 
equations  respectively  of  the  anti-polar  reciprocal  of  L\\pL  =  0. 

(12)  Show,  by  (328)  and  (373),  that  the  condition  that  A 
shall  touch  the  conic  p\^p  =  0,  is  Licf>~^\Li  =  0. 

(13)  Interpret  the  complementary  condition  2h^~^\Pi  =  0. 

(14)  Find  the  locus  of  p  under  the  following  conditions : 
Co,  Bi,  €0,  e,'  are  fixed  points ;  cq  and  e^  are  given  vectors  ; 
p'  =  60'  +  xcq,  p"  =  e/  +  X€i,  and  e^j^'p  =  e^p'^p  =  0. 

Ans.  eoCop  ■  e^p  =  CiCi'p  •  eoCf^p. 

(15)  Write  in  the  result  of  the  last  exercise  Cq"  —  ^o'  for  cq, 
and  Bi'—ei  for  cij  take  the  complementary  equation  and  inter- 
pret it. 

(16)  The  sides  of  a  triangle  cut  the  corresponding  sides  of 
its  polar  triangle  with  reference  to  any  conic  in  three  coUinear 
points.  Eeciprocally,  the  lines  joining  the  corresponding  ver- 
tices have  a  common  point. 


Chap.  IV.]  SCALAR   POINT  EQUATIONS.  153 

(17)  Given  four  points ;  through  them,  two  by  two,  draw 
six  lines,  cutting  each  other  in  three  additional  points,  say  gi, 
q^,  q^;  then  will  any  one  of  the  g-'s  be  the  pole  of  the  line 
through  the  other  two,  with  reference  to  any  conic  through 
the  four  given  points. 

(18)  Derive  the  reciprocal  proposition. 

(19)  Find  the  conditions  that  a  triangle  inscribed  in  a  conic 
shall  have  maximum  area. 

Let  p,  p',  p"  be  the  vertices  of  the  triangle,  subject  to  the 
conditions 

p\<f>P  =  p'l^p' =  p"\^p"  =  0. 

The  area  is  u=pp'p",  and  for  a  maximum  or  minimum  du=0. 

.:  dpp'p"  -f  dp'p"p  +  dp"pp'  =  0. 

But  the  p's  are  independent  of  each  other,  so  that  we  have 

dpp'p"  =  dp'p"p  =  dp"pp'  =  0, 

and  also  <ip\4>P  =  d2)'\<l>p'  =  dp"\cjip"  =  0. 

Hence  p'p"  is  ||  to  \(t>p;  i.e.  to  the  tangent  at  p,  and  similarly 
for  the  other  sides. 

(20)  Show  that,  if  e  and  p  are  any  two  points,  then  e,  ep\e, 
\ep  are  a  normal  system  of  points ;  i.e.  each  point  is  the  anti- 
pole of  the  line  through  the  other  two,  with  reference  to  the 
reciprocating  ellipse. 


Chap.  V.] 


SOLID   GEOilETEY. 


163 


CHAPTER   V 


SOLID  GEOMETRY. 


131.  As  we  are  to  deal  in  this  chapter  with  space  of  three 
dimensious,  the  products  of  two  vectors  and  of  three  points 
will  no  longer  be  scalar ;  but  we  shall  have  instead  — 

The  product  of  three  vectors  a  scalar  quantity,  and 
The  product  oifour  points  a  scalar  quantity. 

Taking  e^,  e^,  e-j,  e^  as  reference  points,  and  letting 

fil  —  ^0  =  £l,   e^  —  fio  =  C2>   ^3  —  fio  =  C3> 

we  shall  assume  always 

1  =  60616363  =  6o€iC2C3  =  Cl«2e3  =  («1  —  ^o)  (62  "  ^o)  (63  "  ^o) 

=  616263  —  636360  +  636061  —  606162  =  I  (60  +  61  +  62  +  63) 
=  4ie,  say. 

Hence  if  p  be  any  point  at  a  finite  distance,  we  shall  have 

i?ci€2C3  =  Pl(6o+6i4-e2  4-e3)  =  4i:>;g  =  l (377) 

The  equations  of  curves  and  surfaces  in  solid  space  may 
appear  under  the  following  forms  : 

Expressed  in  terms  of  points 
Expressed  in  terms  of  planes 
Expressed  in  terms  of  vectors 

Expressed  in  terms  of  points 
Expressed  in  terms  of  planes 
Expressed  in  terms  of  vectors 


Non-scalar  equations. 


Scalar  equations. 


132.  The  non-scalar  equation 

p  =  X(xe)  =  Co  +  2i(x£), 


(378) 


164  DIRECTIONAL  CALCULUS.  [Art.   133. 

€1,  €2,  £3  having  the  values  given  in  the  last  article,  and  x^  being 
eliminated  from  the  third  member  of  the  equation  by  the  rela- 
tion Xo-{-Xi-\-X2  +  Xs=  1,  may  be  called  the  equation  of  solid 
space,  since,  by  giving  proper  values  to  Xq,  Xi,  etc.,  p  may 
become  any  point  whatever  in  space. 

If  a  relation  be  given  between  the  scalar  quantities,  such  as 

fiiXff,  a?!,  0/2,  ajg)  =  0,  or  ji{Xi,  x.2,  Xg)  =  0, 

then  p  will  lose  one  degree  of  freedom  of  motion,  and  will  lie 
on  some  surface.     If  another  relation  be  given,  such  as 

j2{Xo,  Xi,  x-j,  X3)  =  0,  or  j2\Xi,  X2,  Xq)  =  0, 

then  p  will  be  compelled  to  move  on  a  second  surface  simul- 
taneously with  the  first,  and  hence  along  the  curve  of  inter- 
section of  the  two  surfaces.  It  follows  that  the  non-scalar 
equation  of  a  surface  has  two  independent  scalar  variables, 
while  that  of  a  curve  has  only  one.  Eq.  (378)  may  be  written 
in  the  form 

^9  =  6(1 -irec  +  ei-^Vei-f-e.-pje,-!- 63-^:63,  .     .     (379) 

from  which  it  appears  at  once  that  the  scalar  coefficients  are 
proportional  to  volumes  of  the  tetraedra  formed  by  joining  p 
with  the  reference  points.  Since  ^  is  a  unit  point,  the  truth 
of  (377)  appears  also  from  (379).  It  is  easily  seen  likewise 
from  (379)  that  when  p  passes  through  any  face  of  the  refer- 
ence tetrae/dron,  the  coefficient  of  the  opposite  point  changes 
sign.  Thus  when  the  coefficients  are  all  piositive,  p  is  inside 
the  tetraedron ;  when  one  is  negative,  it  has  passed  through  one 
face ;  when  two  are  negative,  it  has  passed  through  two  faces, 
i.e.  through  one  edge;  and  when  three  are  negative,  it  has 
passed  through  three  faces,  i.e.  through  one  vertex. 

133.   If  P=  \p,  then  the  complementary  equation  to  (378) 
or  (379)  is 

P=2o(a;.le)=2o(le-eP),    ...'...     (380) 

and  P  may  be  any  plane  whatever  in  space.  But  if  a  relation 
exists,  as  in  the  last  article,  between  the  scalar  variables,  then 


Chap.  V.]  SOLID   GEOMETRY.  165 

P  moves  according  to  some  definite  law,  and  envelops  a  sur- 
face. 

If  a  second  relation  exists  between  them,  then  P  touches 
two  surfaces  simultaneously,  or  rolls  on  them,  and  therefore 
envelops  a  developable  surface,  which  is  reciprocal  to  a  curve. 

134.  Writing  in  (378)  p  —  ea  =  p,  we  have 

p=2i(a;e), (381) 

a  vector  equation  which,  regarding  p  as  always  drawn  outwards 
from  a  fixed  origin,  represents  a  surface  or  a  curve  under  the 
same  conditions  as  previously  given. 

135.  Equations  of  planes,  lines,  and  points.  If,  together  with 
eq.  (378),  we  have  the  linear  relation 

22(ma;)  =  0, (382) 

then  (378)  represents  a  plane;  for,  since  XQ=p\eQ,  etc.,  we 
have 

So  (mi^le)  =  p12^  Cme)  =  0, (383) 

which  is  the  condition  that  p  shall  be  always  on  the  plane 

I  (moCo  +  Wifii  +  wise,  -f  mgeg) . 
If  we  have  also  the  relation 

^{nx)  =  0, (384) 

p  must  also  lie  on  the  plane  whose  equation  is 

i)|2^(ne)  =  0, (385) 

and  hence  must  lie  on  their  common  line. 

The  equation  of  a  plane  through  any  three  points  pi,  Pi,  Ps 
may  be  written 

p-=Pi  +  x{p2-Pi)  +  y{2h-2h),    ■    •    •     (386) 

of  which  the  scalar  form  is 

i>Pii>2i?3  =  0, (387) 


166 


DIRECTIONAL   CALCULUS. 


[Art.  136. 


which  may  be  derived  from  the  non-scalar  form  by  multiplying 

it  by  PiP^Ps- 

'    The  equation  of  a  plane  through  p^  and  ||  to  Pj  is 

{p-Pi)Pi  =  ^\ (388) 

for  it  is  satisfied  when  2^=  Pi,  and  is  the  condition  that  the 
vector  p  —  x>i  shall  be  ||  to  Pj. 

The  complementary  equations  to  (383),  (385),  and  (387),  viz. : 

P2(TOe)  =  0,  P2(ne)  =  0,  PP^P,Ps  =  0, 

are  the  conditions  that  the  variable  plane  P  shall  always  pass 
through  the  fixed  points  2(wie),  2(ne),  and  PiPjPj,  and  hence 
are  the  plane  equations  of  these  points. 

Eqs.  (233),  (234),  and  (235)  are  the  equations  of  lines  in 
solid  space  as  well  as  in  pilane  space,  though  in  the  former 
they  are  all  non-scalar. 

136.    Vector  equations  of  planes  and  lines.     The  equation 

^l(Ax)=C, (389) 

taken  in  connection  with  (381),  represents  a  plane,  if  p  be 
always  drawn  outward  from  a  fixed  point ;  for,  eliminating  a^ 
between  the  two  equations,  we  have 


or 


{Aap  -  Cea)  (^gCj  -  ^,63)  {A^^  -  -li^s)  =  0 


(390) 


the  non-scalar  and  scalar  forms  of  the  equation  of  a  plane 
through  the  end  of  Cc^  -j-  A^  and  parallel  to  the  vectors 
A^i  —  Ai€^  and  A^t^  —  A^^.     If  a  third  equation, 

'ti(Bx)  =  G\ (3dl) 

be  given,  we  have,  eliminating  x^  and  av. 


/>       «1         C2 

Cl         C2        C3 

=  0,  or 

C   A,  A. 

=  Ai   Ao   A^ 

X- 

C  B,   B, 

Bi    B,   B, 

f  1  J       *2  J     ^^3  ~~  P 

A\i   A^   ^Ag  —  C 
B„  B.^   x,B,-C' 

an  equation  which  evidently  represents  a  right  line. 


(392) 


Chap.  V.]  SOLID   GEOMETRY.  167 

Eqs.  (390)  and  (392)  are  of  the  respective  forms 

p=€  +  Xt'  +  t/c"  ] 

(p-OeV'  =  0      L (393) 

p  =  £  +  2e'  J 

which  are  those  practically  used. 

The  equation  of  a  plane  through  the  ends  of  cj,  e,,  eg  drawn 
out  from  the  origin  may  be  written  at  once  in  either  of  the 
forms 

p  -  £,  =  x(€2  -  £i)  +  y(€s  -  £i)  I 
or  (p_e,)(£,-£,)(£3-£0  =  0      )'      •     •     •     ^       ^ 

the  last  being  equivalent  to 

P(f2e3  +  «3«l  +  £l«2)  =  ci«2e3» (395) 

as  will  be  found  on  multiplying  out. 

137.  Exercises.  —  (1)  What  is  the  meaning  of  the  two 
equations  Fjh  =  0  and  Pj)^  =  0  taken  simultaneously  ? 

(2)  Interpret  the  following  equations  written  in  comple- 
mentary pairs: 

I  Pi3iei£2   =  0  I       f  pPiP2^i        =  0  )       I  pLiLiPi  =  0  I 
\pp,pye  =  0r     XPP,P./i,qoe  =  0i'     XPL,LiPi  =  Or      . 

<pP,Po .  PaAA  =  0  >  U}{k,P,  +  ^-2^2)  =  0  > 

iPpiPi-PsPiPs  =0)'       \P{ky2yi  +  k2P2)  =  0r 

\pU>2  -Pi)  P1P2  =  0 1      \p(k,P,  +  JC2P2  +  hP,)  =  0 1 
1  P(P.,  -  P,)p,2h  =  0  )  '     (  P(lc,p,  +  ^'2i^2  -I-  ^3^3)  =  0  ) 

If,  in  the  fourth  case,  P5  =  Pj  and  p^  =pi,  what  do  the  equBr 
tions  represent? 

(3)  Show  that  p\  (pi  —  4  Ce)  =  0  and  j^lPi  —  C  represent  the 
same  plane,  parallel  to  [  pi  and  distant  from  it  by  the  Amount 
C^  T\p,. 


168  DIRECTIONAL  CALCULUS.  [Art.   137. 

(4)  If  £  be  any  point  at  oo,  show  that  we  have  always 

€\e  =  0 (396) 

From  this  result,  or  otherwise,  show  that 

p\e  =  0 (397) 

is  the  equation  of  the  plane  at  oo. 

(5)  Show  that  the  common  line  of  the  two  planes  P2hP2P3=  0 
and  ppiPiPi  =  C"  is 

(7>2  -Pi)  (PiPiPsPi  -Pi  +  C{Pi  -pi)  - C'ipa  -Pi)). 

(6)  ShoAv  that  the  common  point  of  the  three  planes, 
mP2P  =  C"',  epiPaP  =  C,  epsPiP  =  C"  is 

PiP2PBe  •  e  +C"(e  -2),)  +  C"(e-p,)+C"\e-p,). 

(7)  Show  that  PiP^Pz  =  0  is  the  condition  that  the  three 
planes  p\pi  =i^ii^2  =P\Pi  =  ^  shall  have  a  common  line. 

(8)  Find  the  condition  that  the  plane  p)\j)^  =  0,  together 
with  the  three  planes  of  the  last  exercise,  shall  have  a  common 
point. 

(9)  Show  that  if  the  equations  of  three  planes,  on  being 
multiplied  by  any  constants  and  added,  vanish  identically,  i.e. 
for  all  values  of  p,  then  the  three  planes  pass  through  a  com- 
mon line.  Also  that  if  the  same  holds  for  the  equations  of 
four  planes,  then  the  planes  have  a  common  point. 

(10)  Show  that  when  P\P2PiP\  =  Ci  —  Oj  +  Cg  —  Cy  the  four 

planes     PPlP2Pz=Ci,     PP2p3Pi  =  Oi,      PP3P4Pl  =  C2,     PP4PlP2  =  Cs 

have  a  common  point. 

(11)  Interpret  the  equations  p\€  =  C,  pee'  =  C,  (p  —  e)  |c  =  0, 
(p-ci)(«2-ci)e3=0,     p  =  £+a;|ee' +  ?/££'!£,    p  =  £i+aj(e2-£i) -f-^/fg. 

(12)  Find  the  vector  perpendiculars  from  the  origin  on  the 
planes  of  Ex.  (11). 

Ans.  ^,  ^^'   £,  ^i^2^3- 1(^2-0^3^ 

^'    (££')?'  ((e2-c,)e,y- 

(13)  Find  the  conditions  of  perpendicularity  and  parallelism 
of  the  two  planes  pl£  =  C  and  p|e'  =  C. 


Chap.  V.]  SOLID   GEOMETRY.  169 

(14)  Find  the  vector  perpendicular  from  the  end  of  the 
vector  8  upon  the  line  (p  —  c)  e'  =  0. 

Alls.  £'(e-8)  .[€'. 

(15)  Find  the  conditions  that  the  three  lines  (p  — £)c'  =  0, 
PjPgP  =  0,  P1P2P  =  0  shall  lie  respectively  in  the  three  planes 
p|c"  =  C,  P,p  =  0,  2^\Ps  =  C. 

Ans.  e'le"  =  0  and  e^'  =  C,  P.P^P^  =  0, 
(P-i  —Pi)  \Ps  =  0  and  p^Ips  =  C. 

(16)  Show  that  the  shortest  distance  between  the  two  right 
lines  (p-ci)£2  =  0  and  (p-e,')€2'  =  0  is  (^1 "  ^i>2fg.'.  (Use 
Art.  46.)  ^^^^' 

(17)  Show  that  the  equation  of  the  common  line  of  the  two 
planes   (p  —  e)e'€"  =  0  and  (p  —  ei)e/ti"  =  0  is 

_  ^1^1  ^1       .      /     1       ^^g        .  f  '  4-  Tf'f"  .  f'f" 

P—   I   ,    ,1  ^  ^    I  I  ti  •  fi  -r  ^«  f    •  fi  ci  . 

£  €1  £1  £1  £  €  ' 

(18)  Show  that  the  equation  of  a  plane  through  the  line 
(p  —  £i)£2  =  0  parallel  to  the  line  (p  —  ei')£2*  =  0  may  be  writ- 
ten (p  —  £i)€2|e2C2'  =  0,  and  heiice,  by  Ex.  (17),  that  the  line 

^1  ^2  I  ^2^2  £i£2|e2g2  ,     .         r  , 

P  —     /,  ,  '\2      '^2—    /      ^  fv2    *   ^2    +  a'|e2£2 

\^^2  )-  ^£2^2^"  <^ 

cuts  these  two  lines  at  right  angles. 

(19)  What  are  the  conditions  that  the  pairs  of  right  lines 
q.q.(p  -P.)    =  0  >     ^^  I  (p  -  £,)£,  =  0  )  ^j^^ji  .^^ 

Qiq-liP  -Pi')  =  0  i  (  (p  -  £x')£2'  =  0  ) 

Ans.  pi{q2  -  gi)Pi(g-2'  -  qi)  =  0  and  (e^  -  £i')£2£2'  =  0. 

(20)  Show  that  the  common  point  of  the  three  planes 

ph=Ci,     p\e2=C2,     p\€3  =  Cs 

is  at  the  end  of  the  vector 

(C1C2«3)  ~^  (  Cilc2£3  +  C2IC3C1  +  C3IC1C2)  .        , 


170  DIRECTIONAL  CALCULUS.  [Art.   138. 

(21)  Show  tliat  the  three  planes  of  the  last  exercise  will 
ha\e  a  common  line  if  we  have 

«i«2C3  =  0  and  C1C2C3  +  Cof-zf-i  +  OaCiCa  =  0. 

(22)  Show  that,  if 

C1C2C3C4  —  f-^^iCI\  +  f-s^i^iCo  —  ii^i^iC^  =  0, 
the  plane  ple^  =  C4,  and  the  three  planes  of  Ex.  20  have  a  com- 
mon point. 

(23)  Interpret  the  equations 

f(p\.)  =  0,  f{Tp)=0,  f{Up)  =  0; 
if  /  is  the  symbol  of  a  scalar  function. 

(24)  Three  planes  pass  through  the  three  lines  of  intersec- 
tion respectively  of  the  three  planes 

(p-£i)l«i  =  0,    (p- Co)  1^2  =  0,   and    (p-c3)l€3  =  0, 

each  being  perpendicular  to  the  opposite  plane  (i.e.  the  plane 
through  the  common  line  of  the  first  and  second  planes  is  per- 
pendicular to  the  third,  etc.) ;  find  the  conditions  that  the 
three  first  mentioned  planes  may  intersect  in  a  common  line 
through  the  origin. 

Ans.  ci|c2  •  co^  =  esh  •  ci^  =  cslci  •  C2-- 

(25)  Find  the  shortest  distance  between  the  diagonal  of  a 
cube  and  an  edge  that  it  does  not  meet. 

(26)  Find  the  equation  of  a  line  through  pi  cutting  ij  and  L^. 

Ans.  ppiLiPiL.2  =  0. 

(27)  Derive   (395)  from    (387)    by   transformation,   as  in 
Art.  75. 

138.    TJie  sphere.    The  equation 

p  =  a[(tiC0s6'  +  i2sin6')  sin  ^  +  13  cos  6]  1 
=  a(T  sm  ^  +  13  cos  6),  say,  ) 

so  that 

T  =  iiC0se'  +  i2sin^', (399) 


Chap.  V.]  SOLID  GEOMETRY.  171 

represents  a  sphere.     For,  taking  the  co-square,  we  have 

p?  =  a-(T^  sin-  ^  +  tg-  cos^  6)  =  a^, 

since  t|i3  =  0  and  t^  =  T-  =  1. 

Hence  Tp  =  a  =  constant,  which  is  a  property  of  the  sphere 
with  center  at  the  origin. 

If  the  center  be  at  the  end  of  c,  we  have  for  the  scalar  equa- 
tion T(p  —  e)  =  a ;  whence 

p2  _  Opi^  =  a^  -  £?  or  pl(2e  -  p)  =  c?  -  a".     .     (400) 

This  equation  is  identical  with  (269),  so  that  it  represents  a 
circle  or  sphere  according  as  it  is  interpreted  in  plane  or  solid 
space.  The  properties  of  the  plane-radical  can  be  proved  pre- 
cisely as,  in  Art.  82,  those  of  the  axis-radical  were  demon- 
strated. 

139.  Exercises.  —  (1)  If  a  and  ft  are  any  two  non-parallel 
vectors,  show  that  the  equations 

represent  spheres ;  find  their  centers  and  radii,  and  their  rela- 
tive positions.  If  d  =  Ca  =  Cg  =  0,  show  that  they  cut  each 
other  orthogonally. 

(2)  If  a,  /S,  y,  8  be  any  four  vectors  drawn  outwards  from  a 
point,  and  the  relation 

exists  between  them,  show  that  the  extremities  of  the  four 
vectors  and  their  common  point  all  lie  on  a  sphere. 

(3)  Show  that  the  equations  of  the  tangent  plane  and  nor- 
mal line  to  (400)  are  respectively 

(cr-e)l(p-e)=;i^        (401) 

(<^-«)(p-0  =  0 (402) 


172  DIEECTIONAL  CALCULUS.  [Akt.  140. 

(4)  Show  that  o'  =  p  +  x-^  +  y— ^  is  the  vector  differential 

Clu  (If) 

equation  of  the  tangent  plane  to  (398),  and  from  this  find  the 
equation  of  the  tangent  plane. 

(5)  Find  the  locus  of  the  end  of  p  when  its  distances  from 
two  fixed  points  have  a  constant  ratio  to  each  other. 

(6)  If  p  be  the  vector  radius  of  the  sphere  of  eq.  (400), 
find  the  locus  of  the  end  of  o-,  when  o-  is  subject  to  the  condi- 
tions U(T  =  Up  and  TaTp^Tc^. 

140.    The  paraboloids.     The  equation 

p  =  a;£i  +  ?/c2H cs (403) 

represents  an  elliptic  or  hyperbolic  paraboloid,  according  as  m 
is  positive  or  negative,  as  will  presently  appear.     Let 

ttl  =  IC2«3»    ^2  =  [cgCi,    Og  =  jCjCz,    aud    CjCjC,  =  1  ; 

then  aioaog  =  C2C3  •  C3«i  •  Ci«2  =  (ci^zCs) ^  =  1 » 

Ci|ai  =  1,  Ciloj  =  0,  etc.,  p|aj  =  iC,  ploj  =  y, 
a^  +  my^ 


and 


Pl'^3=       « 


whence        {p\<x^'^ -^in(^p\a^'^  =  ap\a^  =  p\<^p-^   ....     (404) 
in  which       ^p  =  aj  •  p|ai  +  w^a  •  pl^a (405) 

Eq.  (404)  is  the  scalar  form  corresponding  to  (403). 
Find  the  intersection  of  this  locus  with  the  plane 

pl(»hai  +  n2a2)  =  0 (406) 

Eliminating  p|ai,  we  obtain 

(''W'=;;?Ti^-^i°' w 

Eqs.  (407)  and  (406)  taken  simultaneously  represent  a 
parabola,  as  appears  at  once  from  the  results  of  Chap.  III. ; 
thus  any  plane  through  CoCg  cut»  a  parabola  from  the  surface, 
Co  being  the  origin. 


Chap.  V.]  SOLID   GEOMETRY.  178 

Again,  intersect  the  surface  by  the  plane 

Pla3  =  c; (408) 

whence        p\<lip  =  (p\ai)- -^m(p\a.2y  =  ac (409) 

Eqs.  (409)  and  (408)  taken  together  represent,  by  Chap. 
III.,  an  ellipse  or  hyperbola  according  as  m  is  positive  or  nega- 
tive ;  thus  all  sections  parallel  to  cic^  are  ellipses,  or  else  all  are 
hyperbolas ;  hence  the  name  of  the  surface  as  stated  at  the 
beginning  of  this  article. 

141.   Eq.  (404)  may  be  written  in  the  form 


pj(ai  +  a2V— m)  .pKai-aaV— m)  =  apla3| 
or         p\l3i-p\(S2  =  ap\as  ]' 

which  shows  that  the  surface  passes  through  the  two  common 
lines  of  the  plane  p\as  =  0,  and  the  two  planes  p|/?i  =  0  and 
pjyg,  =  0.  These  are  real  when  m  is  negative,  and  imaginai'y 
when  m  is  positive. 

Now  (410)  will  be  satisfied  by  any  value  of  p  which  satisfies 
simultaneously  either  of  the  following  pairs  of  equations,  viz.  • 

\p\ft,  =  lp\a,l^      ^pl^,  =  ^^p\aJ ^^^^^ 

(p!A  =  n  )  (p\Pi  =  n'  ) 
Each  of  these  pairs  represents  a  right  line,  which  will  change 
its  position  as  we  change  n  or  n' :  hence  it  appears  that,  when 
m  is  negative,  there  are  two  systems  of  right  lines  that  lie 
wholly  on  the  surface ;  these  are  called  two  systems  of  recti- 
linear generators  of  the  surface.  They  are  imaginary  when  m 
is  positive.  By  Ex.  (9)  or  (22)  of  Art.  137  it  is  easily  shown 
that  no  two  generators  of  the  same  system  intersect,  while 
every  generator  of  one  system  cuts  every  one  of  the  other. 

142.   Exercises.  —  (1)  Find  the  common  point  of  two  gen- 
erators belonging  to  the  systems  n  and  n'  respectively. 

Ans. [aV— w(w+7i')ci+a(w'— w)«2+2nn  cgV— m]. 


174  DIRECTIONAL  CALCULUS.  [Art.  143. 

(2)  Show  that  all  the  generators  of  either  system  of  a 
hyperbolic  paraboloid  are  parallel  to  a  single  plane. 

(3)  Find  the  equations  of  the  tangent  planes  to  the  loci  of 
eqs.  (403)  and  (404)  ;  and  show  that,  when  m  is  negative,  the 
tangent  plane  contains  a  generator  of  each  system. 

(4)  If  ai,  02,  ag  in  (404)  form  a  unit  normal  system,  so  that 
we  may  write  ai  =  ci  =  i^  etc.,  find  the  locus  of  the  points  on 
the  surface  at  which  the  two  generators  passing  through  each 
of  such  points  are  perpendicular  to  each  other. 

Ans. 

a^(l+m)r_^^^_«/  _^_^V      4mn     / t^\l 

8m      1_  7i\       ^-mj     a(l+m)\       V-m/J 

n  being  variable.     This  equation  represents  a  hyperbola. 

2 

(5)  In  eq.  (404)  change  the  origin  by  putting  p'  -] —  eg  for 

p,  and  then  determine  the  nature  of  the  surface  when  a  =  0. 
Also  when  a  =  c  =  0. 

(6)  Taking  ai,  a2,  og  as  in  Ex.  (4),  show  thQ±  p\e=  C  Avill 
represent  a  plane  tangent  to  the  locus  of  (404)  if  we  have 

C=  —  a  •  e|«^~^e  -f-  4c|c3. 

143.  The  central  quadrics.  The  surfaces  represented  by  the 
three  equations 

p  =  T  sin  $  +  ctg  cos  0 

p  =  T  cosec  6-{*CLs  cot  0    >  , (412) 

P  =  t'  cosec  6  +  cig  cot  6 

in  which        t  =  aij  cos  6'  +  ftij  sin  6'  ^ 

T'=atisec^'  +  &t2tan^'|' ^       ^ 

are  called  respectively  the  ellipsoid,  the  hyperholoid  of  one  sheet, 
and  the  hyperholoid  of  two  sheets,  for  reasons  which  will  pres- 
ently appear. 


Chap.  V.]         ^  SOLID  GEOMETRY.  175 

By  comparison  with  eqs.  (276)  and  (277)  it  will  be  seen  that 
eqs.  (413)  represent  respectively  an  ellipse  whose  semi-axes 
are  a  and  b,  and  a  hyperbola  with  the  same  semi-axes,  each  ' 
lying  in  the  plane  tji.,,  the  vector  radius  of  one  curve  being  t, 
and  that  of  the  other  t'.  In  the  first  of  eqs.  (412)  give  to  t 
some  particular  value  consistent  with  (413)  ;  then  the  equation 
represents  an  ellipse  whose  semi-axes  are  Tt  and  c ;  hence  the 
first  of  (412)  represents  a  surface  generated  by  an  ellipse  re- 
volving about  ig  as  an  axis,  c  being  the  semi-axis  along  tg,  while 
the  other  semi-axis  is  the  radius-vector  of  the  ellipse  in  11I2 
whose  semi-axes  are  a  and  b. 

Similarly,  the  second  of  (412)  represents,  for  any  given  value 
of  T,  a  hyperbola  whose  semi-axes  are  Tt  and  c,  and,  when  t 
varies  subject  to  ^(413),  it  represents  the  surface  generated  by 
this  hyperbola  revolving  about  13,  having  c  for  its  semi-axis 
along  13,  while  the  other  is  the  radius  vector  of  the  ellipse  in 

Finally,  the  third  of  (412),  for  any  given  value  of  t',  repre- 
sents a  hyperbola  with  semi-axes  Tt'  and  c,  t'  being,  as  we  saw 
above,  the  radius-vector  of  a  hyperbola  in  ijia.  Thus  the  sur- 
face is  generated  in  this  case  by  a  hyperbola  revolving  about 
13,  having  its  c-axis  constant,  while  its  other  semi-axis  is  the 
radius-vector  of  a  hyperbola  in  tjig  whose  semi-axes  are  a 
and  b. 

The  methods  of  generation  of  these  three  surfaces  show  that 
the  first  is  a  limited  surface,  having  no  real  points  at  00,  and 
hence  that  no  plane  can  cut  from  it  a  hyperbola  or  parabola ; 
hence  the  name  ellipsoid :  that  the  second  is  generated  by  a 
hyperbola  in  such  a  way  as  to  form  one  continuous  surface,  so 
that  any  two  points  whatever  lying  on  it  can  be  joined  by  a 
line  also  lying  wholly  on  it-,  hence  the  name  hyperboloid  of 
one  sheet :  finally,  that  the  third  is  generated  by  a  hyperbola 
in  such  a  way  as  to  form  two  distinct  portions,  or  sheets,  so 
that  points  on  these  respective  portions  cannot  be  joined  by  a 
line  lying  wholly  on  the  surface ;  whence  the  name  as  given 
above. 


176  DIRECTIONAL  CALCtTLUS.  [Art.   144. 

144.   Eliminating  6  and  6'  from  eqs.  (412),  we  obtain 

/1\0  /     ]      \  9  /     ]      \  9 


(t) 


'^i+'t  +T>=i 


(414) 


the  scalar  equations  of  the  same  surfaces. 

As  an  exercise  let  the  student  determine  the  nature  of  the 
sections  of  the  surfaces  of  (414)  by  the  three  planes  pjii  =  Cj, 
p\i2  =  C2,  p\l3  =  C3  when  Ci  >  a,  (72  >  b,  Co  >c,  Ci  —  a,  Cg  =  b, 
C3  =  e,  C\  <a,  Co<  b,  Cs<c. 

145.  Write 

.        h:m^'ll4^±'JLlP\^-,      ....     (415) 
a^  b-  & 

then  the  equation 

p\<i>p  =  l  .    ■ (416) 

is  equivalent  to  any  one  of  the  equations  (414),  if  we  select 
the  signs  properly  in  (415). 

The  function  ^  as  given  in  (415)  is  evidently  self-conjugate, 
and  possesses  all  the  properties  proved  in  Arts.  86  and  89. 
Eq.  (416)  being  of  precisely  the  same  form  as  (282),  any 
operations  performed  on  (282)  which  did  not  depend  on  the 
form  of  <^  will  give  identical  results  when  performed  on  (416), 
the  interpretation  of  the  results  being,  of  course,  different. 
The  discussions  which  follow  hold  for  any  linear^  vector^  self- 
conjugate  form  of  <j},  except  when  <^  is  specially  restricted  to  the 
form  (415). 

146.  Tangent  plane  and  normal.  By  differentiation  we  have, 
precisely  as  in  Art.  87,  that  ^p  is  parallel  to  the  normal  at  the 
end  of  p,  and  hence  that  the  equations  of  the  tangent  plane 
and  normal  line  at  the  end  of  p  are  respectively 

(r|<^p  =  l (417) 

and  {iT-p)^p  =  (i, (418) 


Chap.  V.]  SOLID   GEOMETKY,  177 

which  are  identical  in  form  with  (284)  and  (285).  Also,  as 
in  the  same  article  the  perpendicular  from  the  origin  on  the 
tangent  plane  is 

7~-^^P (419) 

147.  Diametral  plane.  Repeating  exactly  the  operations  of 
Art.  88,  we  obtain  the  same  equation 

o-|«^c  =  0, (420) 

which  now  represents  a  plane,  the  locus  of  the  middle  points  of 
a  system  of  chords  parallel  to  c.  This  plane  is  perpendicular 
to  <f>e,  i.e.  parallel  to  the  tangent  planes  at  the  ends  of  the 
diameter  parallel  to  e,  and  is  said  to  be  conjugate  in  direction 
to  e.  Also,  any  straight  line  in  this  plane  is  conjugate  in 
direction  to  c,  so  that  (420)  is  the  condition  that  two  vectors 
a-  and  e  shall  be  conjugate.  If  vectors  o-i  and  o-s  be  taken  in 
(420)  so  that  the  tangent  plane  at  the  point  where  0-2  drawn 
out  from  the  origin  pierces  the  surface  is  parallel  to  o-j,  then 
we  shall  have 

(ri\<t>(T2  =  0  =  a-i\<f>€  =  o-2|<^c, 

and  <ri,  0-3,  and  c  form  a  set  of  conjugate  directions.  If  a,  ^,  y 
are  three  vectors  which  satisfy  the  conditions  just  found,  and 
also  the  equation  of  the  surface  p\cf)p  =  l]  that  is, 

a|<^;3  = /8I«^y  =  y|(^  =  0 

al^a  =  ^|<^^  =  y\<f)y 
then  a,  yS,  y  are  a  set  of  conjugate  semi-diameters.    From  these 
relations  we  have  a  =  m\(f>(3<f>y  ;  multiply  into  \<fxi. 
.'.   a|^  =  1  =  7n(f>a(f)ft<f>y. 

Substitute  value  of  m,  and  we  have 
a  •  (f)a(j>P(f)y  =  \4*P^y  1 

and,  similarly,        y8-</>a<^;8</)y  =  |<^y<^a   > (422) 

y  •  <^a<^/3</>y  =  |<^a^/3  J 
In  the  same  way  we  find 

<^a.a/3y  =  I;8y,     <^/3  •  a/3y  =  |ya,    <f>y  ■  a^y  =  \a^.  (423) 


=  0) 

^J;  ....    (421) 


178  DIRECTIONAL,  CALCULUS.  [Art.  148. 

148.  Interpretation  of  the  equation  (r|^e  =  l.  In  the  first 
place  it  evidently  represents  some  plane  parallel  to  that  of 
(420),  i.e.  conjugate  in  direction  to  c. 

If  we  have  £|^£  =  1,  the  equation  is  identical  with  (417)  and 
hence  represents  the  tangent  plane  at  the  end  of  c.  In  gen- 
eral, let  p  be  the  vector  to  the  point  of  contact  of  some  plane, 
passing  through  the  end  of  e  and  tangent  to  the  surface  :  then 
the  equation  of  this  tangent  plane  Avill  be  o-|^/3  =  l,  with  the 
condition  c|<^/3  =  1  =  p|<^e,  which  makes  the  plane  pass  through 
the  end  of  e.  If  p  be  variable  in  the  last  equation,  we  have  the 
locus  of  the  end  of  p,  and  the  equation  becomes  identical  with 
that  at  the  head  of  the  article  when  we  put  o-  for  p.  Hence 
the  equation 

o-I</)£  =  l (423a) 

represents  a  plane  containing  the  points  of  contact  of  all  tan- 
gent planes  to  the  surface  which  pass  through  the  end  of  e,  or, 
in  other  words,  the  plane  of  the  curve  of  contact  of  the  cir- 
cumscribed cone  whose  vertex  is  at  the  end  of  c.  Of  course 
the  end  of  e  may  be  so  situated  that  this  cone  is  imaginary. 
The  plane  o-j^e  =  1  is  called  the  polar  plane  of  the  point  at 
the  end  of  e,  while  this  point  is  the  pole  of  that  jolane. 

Now  of  the  infinite  number  of  directions  of  <t  in  (423  a)  one 
will  evidently  coincide  with  that  of  c ;  when  o-  has  this  direc- 
tion suppose  it  to  become  fixed  and  c  to  vary ;  we  shall  then 
have  the  plane  c](^o-  =  l  parallel  to  the  plane  which  we  had 
when  e  was  fixed  and  o-  varied,  and,  since  one  value  of  e  must 
be  its  original  value,  the  plane  must  now  pass  through  the  end 
of  c  in  its  original  position  ;  thus  the  positions  of  the  pole  and 
polar  plane  have  been  exchanged  as  regards  their  distance 
from  the  center  measured  in  the  direction  e. 

If  a  point  pi  he  on  the  polar  plane  of  p.,,  then  loill  p).,  he  also  on 
the  polar  plane  ofp^. 

Let  pi  and  p^  be  at  the  ends  of  c^  and  co  respectively ;  the 
polar  planes  are  then  o-jt^ti  =  1  and  o-|^£2  =  1  respectively ;  if 
Pi  be  on  the  polar  plane  of  p)^,  we  must  have  the  equation  satis- 


Chap.  V.] 


SOLID    GEOMETRY. 


179 


fied  when  <t  =  ci;   i.e.  ei\(f>€2  =  1  =  €2\<f>€i ;  but  this  is  also  the 
condition  that  pa  shall  be  on  the  polar  plane  of  p^. 

Finally,  we  have,  precisely  as  in  Art.  92,  that  the  semi- 
diameter  along  e  is  a  mean  proportional  between  the  distances 
along  e  to  the  point  and  to  its  j^olar  pdcme. 


L* 


149.  As  in  Art.  1)1,  we  have 

p\(fip  =  p\<i>''ft>-p  =  <l>-p\<l>-p  =  (^V)"^  =  1? 

or  r</>2p  =  l; (424) 

so  that,  when  p  is  a  vector  of  the  surface,  (f>-p  is  always  a  unit 
vector.     Also,  if  a,  /3,  y  are  vector,  conjugate  semi-diameters, 

and,  similarly,    ff)^^\cf>''y  =  ^-y|^^a  =  0, 

so  that  <^-a,  ff>-j3,  <ji'-y  form  a,  U7iit,  normal  system  of  vectors. 

150.  The  volume  of  the  parallelopiped  formed  by  tangent 
planes  at  the  ends  of  a  set  of  conjugate  diaineters  of  an  ellipsoid 
is  constant. 

Taking  a,  y8,  y  as  in  the  last  article,  the  required  volume  is 
8a/3y.  Now  with  <^,  as  in  eq.  (415)  with  the  upper  signs,  we 
have 

4>~  p  =  CUi  •  p\Li  +  i>L2  •  p\l.2  -f-  CI3  •  p\ts, 

_i.    1  1  1  .     .    1 

and  a  =  <f>  '(fy^a  =  aii  •  t]l<ji)-a  +  bu  •  i2\4>  a  -\-  Ci^  •  igl^-a 

y8  =  <f--cfj^~l3=  ai, '  ii\cj>^^+etG. 


y  =  (f>  -(f>-y  =  ail  •  ti|^^y  +  etc. 
iil^^a,   i2\<l>-a,  63] <^  a 


(425) 


8al3y  =  8abc 


i,\cf>^(S,  L2\^^^,  i3l«^^ySJ 


=  8  abc  •  iiWs  •  (f>-a<f}'fi<f>-y 
=  8  abc, 


'il^^y?  h\<i>'y,  t3l<^"y 
by  the  last  article  and  equation  (195) . 

Take  the  co-square  of  each  of  equations  (425)  and  add  them, 

and  we  have 

a?  +  /6^  +  y'  =  ci'  +  &'4-c^ (426) 


180 


DIRECTIONAL.  CALCULUS. 


[Art.  151. 


151.   The  equation 

{fiypr±{yapy±ia/3py-  =  iaPyy    ....      (427) 

represents  a  central  quadric  referred  to  the  conjugate  semi- 
diameters  a,  yS,  y.  The  origin  is  at  the  center  because  the 
equation  is  unchanged  by  putting  —  p  for  +  p.  If  a  =  aij, 
^  =  6i2)  y  =  C43,  the  equation  becomes  identical  with  (414) 
Eq.  (427)  is  satisfied  when  p=a,  ov  p=^^ ±  1,  or  p=yV±T, 
BO  that,  when  the  signs  are  all  positive,  the  ends  of  a,  /3,  y  are 
real  points  on  the  surface;  when  one  sign  is  negative,  one 
point  becomes  imaginary ;  and  when  two  signs  are  negative, 
two  points  become  imaginary. 
If  we  write 


(a^y) 

eq.  (427)  becomes  p\<f>p  =  1,  and  we  have  at  once 
the  conditions  for  conjugate  directions. 


152.   Rectilinear  generators. 
the  form 


Eq.   (414)  may  be  written  in 


+ 


.6V±1      cVtI 


6V±1 


=    1  + 


;Vt1 

a 


\,.     (429) 


or         a^p\a  •  p\a'  =  (a  +  p|ii)  (a  —  pjii) 


which  shows  that  the  surface  passes  through  the  four  lines  of 
intersection  of  the  pair  of  planes  p\a  =  0  and  pa'  =  0  with  the 
pair  of  planes  p\ii  =  —  a  and  p\ii  =  a.  Now  if  we  take  the 
upper  signs  throughout,  a  and  a'  are  imaginary  ;  hence  for  the 
ellipsoid  these  lines  are  imaginary.  Again,  taking  the  loicer 
signs  throughout,  a  and  a'  are  imaginary,  and  the  lines  are 
therefore  imaginary  for  the  hyperboloid  of  two  sheets.  If, 
however,  we  take  the  upper  sign  for  the  second  term  of  (415) 


Chap.  V.]  SOLID   GEOMETRY.  181 

and  the  lower  sign  for  the  third  term,  or  vice  versd,  then  a  and 
a'  are  real,  and  the  four  lines  of  intersection  of  the  two  pairs 
of  planes  are  real  lines  on  the  surface,  which  is  now  a  hyper- 
boloid  of  one  sheet.     In  this  case  we  have 

a  =  b~h.2  4-  c~^3  and  a'  =  6~^i2  —  c~%. 

Either  of  the  two  pairs  of  planes, 

p\(aa  —  ?Ui)  =  ?ia")  {p\(aa  +mti)  =  ma 

and   ]    \^     ,      1  a    }■,      (429a) 


(  p\{aa  —nLi)  =  na\  (P.v'^  ■tmLi)  = 

^  n         n    ■'  ^  m 


m 


taken  simultaneously  satisfies  eq.  (429).  Hence  the  two  planes 
of  each  pair  intersect  in  a  line  lying  wholly  on  the  surface. 
By  varying  m  and  n  we  thus  obtain  two  systems  of  rectilinear 
generators  of  the  hyperboloid  of  one  sheet. 

153.  Exercises.  —  (1)  Show,  by  Ex.  22  or  9  of  Art.  137, 
that  no  two  generators  belonging  to  the  same  system  intersect, 
while  every  generator  of  one  system  cuts  every  generator  of 
the  other. 

(2)  Show  that  the  vector  to  the  common  point  of  a  genera- 
tor of  the  system  n  and  one  of  the  system  m  is 

p  =  (m-\-  ri)~^[aii(m  —  w)  +  hi^(mn  + 1)  -f  CL^{inn  —  1)]. 

(3)  Eind  the  condition  between  m  and  n  in  order  that  a 
generator  of  one  system  may  be  perpendicular  to  one  of  the 
other. 

Ans.  4 a-  —  Z>- f m]i 7i]  —  <r[  — |-  ??i  ](  -  +  ?i )  =  0. 

\m     ■    J\ii        J         \m         j\n        J 

(4)  Show  that  the  projections  of  the  generators  on  the  re- 
ference planes  are  tangent  to  the  principal  sections  of  the 
surface ;  that  is,  that  the  projecting  planes  of  the  generators 
touch  the  surface  at  points  lying  in  the  reference  planes. 

(5)  Show  that  if  a,  ji,  y  be  substituted  for  aq,  612,  0%  in 
(412)  and  (413),  those  equations  will  be  equivalent  to  (427). 


182  DIRECTIONAL   CALCULUS.  [Art.  15:3. 

(6)  Find  the  locus  of  the  intersection  of  tangent  planes  at 
the  ends  of  conjugate  semi-diameters  a,  f3,  y. 

Let  a  be  the  vector  to  the  point  of  intersection ;  then 

o-  =  a  +  ;8  +  y  and  a\<}>(r  =  3 
is  the  equation  of  the  locus,  a  similar  surface. 

(7)  Show  that  the  locus  of  the  extremity  of  the  vector  ff>p 
is  <r|^~'o-  =  1,  if  o-  =  <j>p. 

(8)  Show  that,  when  T<}>p  =  k  =  const.,  the  locus  of  the  end 
of  p  is  /3|<^V  =  ^'^• 

(9)  Find  the  equation  of  the  pedal  surface  of  the  central 
quadric ;  i.e.  the  locus  of  the  foot  of  the  perpendicular  from 
any  point  upon  the  tangent  plane. 

If  o-  be  the  vector  to  a  point  of  the  tangent  plane,  and  e  the 
vector  to  the  fixed  point,  we  have 

o-  —  £  =  a;0p,  p\cf}p  =  1,  and  crl^p  =  1  ; 

therefore  o-| (o-  —  e)  =  a;  and  (or  —  e)  |^~^ (o-  —  e)  =  xr, 

whence  (o-- £)|<^-\(r- e)=[o-|(o- -  e)]-,     .     .     (430) 

a  surface  of  the  fourth  order.     If  we  change  the  origin  to  the 
end  of  c,  by  writing  p  for  o-  —  «,  we  have 

p\<l>-'p  =  [p\{p  +  e)J- ^    .      (431) 

(10)  Show  that  the  vectors  joining  any  point  on  the  surface 
with  the  extremities  of  a  diameter  are  conjugate  in  direction. 

(11)  Find  the  value  of  C  so  that  pjc  =  C  may  represent  a 
plane  tangent  to  p\<f>p  =  1. 

Ans.  C  =  V^1^^ 

Compare  with  (417),  and  note  that  the  equation 
is  independent  of  Tt. 


Chap.  V.]  SOLID   GEOMETRY,  183 

(12)  Taking  <^  as  in  (Ho),  and  cj,  €2,  eg,  as  unit  normal  vec- 
tors, show  that 

£i|<^ei  +  £2|<^t2  +  csl^cg  =  —  ±  —  ±  — 
CI2       O2       C2 

(13)  "Find  by  Exs.  (11)  and  (12)  the  locus  of  the  common 
point  of  three  perpendicular  tangent  planes  to  a  central 
quadric. 

Ans.  a^  =  a^  ±b'^  ±  c^. 

154.  Condition  that  p\(f)p  shall  be  factorable.  Proceeding  pre- 
cisely as  in  Art.  114,  in  fact  simply  putting  vectors  for  points 
in  that  article,  we  have  for  the  required  condition 

<j}\<}>fji<f>v  =  0, (432) 

X,  IX,  V  being  any  three  vectors  whatever. 

The  function  p|<^p  —  fcp-  is  always  factorable  ;  for,  writing  it 
in  the  form  p\{<^  —  k)p,  and  putting  (}>  —  k  for  <}>  in "(432),  we 
have 

(cf>-Jc)\((l>-Jc)ti((l>-k)v  =  0,  .     .    .     (433) 

a  cubic  in  A;  which  must  have  at  least  one  real  root :  hence  a 
value  of  k  can  always  be  found  which  will  make  p\((t)  —  k)p 
the  product  of  two  linear  factors. 

155.  The  {j>  function  in  general.  Any  linear  vector  function 
may  be  written  in  the  form 

<^p  =  £1  •  ci'l/3  +  £2  •  £2'ip  +  C3  •  ^s'Ip-  •  •  •  (434) 
This  may  be  shown  precisely  as  was  done  in  Art.  97  for  two- 
dimensional  space.     The  function 

<f>,p  =  e/  •  £iIp  +  £2'  •  C2IP  +  £3'  •  ^s\p   •     •     •     (435) 

is  conjugate  to  (ft;  i.e.  a\(f>p  =  p\(f>^(r,  and  (^-t-<^<.)/>  is  always 
se?/-con jugate,  as  was  shown  in  Art.  97.     Again, 

p\^P  =  pl't'cP  or  p\  (<^  —  <li,)p  =  0. 

••.  (^  —  (fic)p  is  perpendicular  to  p,  and  we  may  write 

(</.-«^,)p=l£p, (436) 


184  DIRECTIONAL  CALCULUS.  [Art.  156. 

e  being  some  real  vector  when  <f>  is  not  self -con  jugate.     Then 
we  have 

<f>P  =  H4>  +  <t>c)p  +  Hi>-<f>c)p==U'f>  +  ^c)p  +  ¥<P-     (437) 

In  Art.  129  replace  2^  by  p,  e  by  e,  cji,  q^,  fh  by  X,  fi,  v,  and 
we  have  the  following  inversion  formulae : 

(f>^<f>^<l>^v '  ^"V"  =  V"  •  HcP-i>cV (439) 

Putting  also  m's  for  k's  and  gr  for  n,  Ave  have  from  (374), 
(375),  and  (376), 

wio^"  V  =  (*^i  ~"  '''^2'A  +  ^^)P) (440) 

(mo  +  miQf  +  w^gr^  +  9^)  (<^  +  ^)  "  V       |  .^^-^x 

=  mo<t>-^P  +  g(m._-<}i)p-{-g'p,) 

{cf>^  —  m2<l>-  +  mi(f>-mo)p  =  0 (442) 

The  coefficients  wio,  mj,  m2  are  invariants;  i.e.  their  value  is 
the  same  whatever  X,  /x,  v  may  be,  which  may  be  shown,  as  in 
Art.  98. 

156.  The  general  scalar  equation  of  the  second  degree  in  terms 
of  vectors.     This  is  identical  in  form  with  eq.  (312)  for  plane 

space,  viz. : 

p|<^p  +  2y|p=C; (443) 

for  all  second-degree  terms  may  be  included  in  pi<^p,  and  all 
first-degree  terms  in  2  y\p. 

We  will  first  show  that  the  surface  represented  by  (443)  has, 
in  general,  cyclic  sections;  i.e.  that  certain  planes  will  cut 
circles  from  the  surface.  Add  and  subtract  kf^  to  and  from 
eq.  (443),  which  thus  becomes 

kf^^2y\p-C+p\{<i>^k)p  =  0. 

By  Art.  154,  k  can  be  so  determined  that  p|(<^  —  k)p  shall  be 


Chap.  V.]  SOLID  GEOMETRY.  185 

factorable ;  i.e.  it  may  be  written  p|a  •  p\a.',  k  being  one  of  the 
roots  of  (443).     Hence  (443)  may  be  written 

kf^  +  2y\p-C  +  p\a.p\a'  =  0,  .     .     .     .      (444) 

which  represents  a  surface  passing  through  the  curves  of  in- 
tersection of  the  sphere  kf^  -\-2y\p  —C=0  with  the  two  planes 
p|a  =  0  and  p\a'  =  0.  As  these  curves  are  necessarily  circles, 
these  two  planes  cut  circles  from  the  surface. 

157.  Let  us  apply  the  results  of  the  last  article  to   eq. 
(414).     Eq.  (433)  becomes  in  this  case,  if  we  put  A  =  ij,  /a  =  ij, 

(a---k)(±b-''-k)(±c-^-k)  =  0, 

which  gives  the  three  values  of  k.  Using  the  value  k  =  b~^  for 
the  ellipsoid  first,  we  have 

6-  b^ 

or,  writing  —  —  1  =  Ci^  and  1 =  Ca^,  and  reducing, 

<r  a 

or  again      p' (dig  +  ?^i)  •  p\  {^ih  —  ^af-i)  =  &^  —  p-,    •     •     •     (445) 
an  equation  in  the  form  of  (444),     The  two  planes 
p!(eii3  4-  Cgti)  =  0  and  pKcjig  -  egii)  =  0 

cut  cyclic  sections  from  the  ellipsoid ;  they  pass  through  the 
6  axis  of  the  ellipsoid,  and  are  inclined  to  the  a  axis  at  the 

angles  tan~^  -?  and  tan~M  — ? ) ;  i.e. 

tan-^O^^^  and  tan-^-^J^^-   ' 

The  values  of  Ci  and  Ca  are  real  only  when  6  lies  between  a  and 
c  in  value. 

For  the  hyperboloid  of  two  sheets  b^  and  c^  are  negative,  and 
therefore  Ci  and  €2  are  real  when  6  lies  between  a  and  c  in  numer- 


186  DIEECTIONAL   CALCULUS.  [Art.  158. 

ical  value,  so  that  the  cyclic  planes  of  (445)  pass  through  the 
greater  axis  which  does  not  pierce  the  surface. 

For  the  hyperboloid  of  one  sheet  let  a-  be  negative  and  h^ 
and  c^  positive ;  then  Cs  is  real,  and  d  is  real  when  b  is  greater 
than  c,  so  that  the  cyclic  planes  of  (445)  pass  through  the 
greater  axis  which  pierces  the  surface. 

158.   We  will  next  apply  the   results   of  Art.   156  to  eq. 
(404),  substituting  ij  and  ig  for  ai  and  02- 
We  have  then  from  (433) 

{l-k)(m-k)k  =  0. 

The  root  ^  =  0  shows,  as  is  evidently  true,  that  p\(fip  is,  in 
this  case,  factorable  without  any  addition  of  a  multiple  of  p-. 
This  leads  to  the  rectilinear  generators  of  the  hyperbolic 
paraboloid  as  in  Art.  141,  which  are  sections  of  the  surface  by 
infinite  spheres.  Taking  next  the  root  k  =  m,  we  have,  after 
reduction,  the  equation 


p\  (ii Vl  —  m  +  isVm)  •  p\  (ii  Vl  —  m  — 13 Vm) 
=  ap|i3  —  mpr 


|,      .     (446) 


which  gives  real  cyclic  sections  when  m  is  positive  and  less 
than  unity.     Similarly,  the  root  k  =  l  gives 


p\  (t2  Vm  —  1  + 13)  .  p|  (i2  Vm  —  1  —  tg)  =  ap\is  —p\     .     (447) 

which  gives  real  cyclic  sections  when  m  is  positive  and  greater 
than  unity.  Eqs.  (446)  and  (447)  both  represent  elliptic 
paraboloids,  so  that  the  only  cyclic  sections  of  the  hyperbolic 
paraboloid  are  the  generators  as  mentioned  above. 

159.  Exercises.  —  (1)  Show  that  the  two  planes 
p\  (eii3  +  egti)  =  C  and  p\  (eii3  —  Csm)  =  -  C 
cut  the  surface  p|<^p  =  1  in  circles  that  lie  also  on  the  sphere 
p^-\-2C^^\i,  =  b--C\ 


Chap.  V..]  SOLID  GEOMETRY.  187 

(2)   Show  that,  when  C  =  ±  Va^  —  c^,  the  planes  of  the  last 
exercise  touch  the  surface  in  the  points 


p  =  (eiC%  ±  Zsa\)  ^  ( ±  Va^  —  c-). 
These  points  are  called  the  umbilici. 

(3)  Show  that  the  locus  of  the  common  point  of  tangent 
planes  at  the  ends  of  three  perpendicular  radii  vectores  is 

Let  the  vector  radii  be  pi,  p2,  ps ;  then 

Pi\(f)pi  =  1,  etc.,  or\(fipi  =  1,  etc.,  and  pjlpa  =  palpa  =  Pslpi  =  0. 

Hence, 

Now,  by  (178),  putting  <l>pi  for  cj,  etc.,  we  have 

O"  =  {^Pl^P2fl>P3)~W^P\<f>P2  +  \<l>P2i*Ps  +  |«AP3^Pl)> 

or,  by  (439),      =  (pipips)  '^<t>~'^  ( \p1p2  +  I/02P3  +  Ipspi) 

=  (Tp,Tp,Tp,)-'r'(^^  •  P3  +  etc.) 

V/T       P2-       P3- 

^^  ^         '^        py  pi     pi-     a'     b'     c'' 

by  !Ex.  12,  Art.  153,  since  pi\(f>pi  =  1  gives  — -  =  Upi\<fiUpi,  etc. 

Pr 

(4)  Find  the  equation  of  a  cone  with  vertex  at  the  end  of  c 
circumscribed  about  the  central  quadric. 

Ans.  With  the  origin  at  the  vertex,  the  equation  is 
p\4>p  —  ^p\4>^4'P  =^  ^* 

(5)  Show  that  the  vector  to  the  pole  of  the  plane  (r|a  =  C 
is</>ia-H(7. 


188  DIRECTIONAL  CALCtJLtJS.  [Art.  160. 

(6)  A  plane  is  tangent  to  a  central  quadric ;  find  the  locus 
of  its  pole  with  reference  to  another  quadric  concentric  with 
the  first. 

Ans.  If  the  equations  of  the  quadrics  are  p\4>2p  =  1  and 
p\<pip  =  1  respectively,  the  locus  is  p|^2~Vi'P  =  1- 

(7)  Show  that  any  sphere  with  its  center  at  the  origin  cuts 
the  surface  p\ii  •  pji,  +  p\h  •  p\h  +  plh '  P|M  =  1  in  circles  lying  in 
two  parallel  planes. 

(8)  Show  that  the  principal  sections  of  the  surfaces  obtained 
by  giving  different  values  to  k  in  the  equation 

p\(r'-Jc)'p  =  l (448) 

have  the  same  foci,  <f>  being,  as  in  (415),  with  the  upper  signs. 
These  surfaces  are  said  to  be  confocal. 

(9)  Show  that  through  any  point  in  space  three  confocal 
surfaces  pass,  corresponding  to  three  values  of  k,  and  that 
these  surfaces  cut  each  other  at  right  angles. 

See  Ex.  2,  Art.  93. 

(10)  Show  that  the  equation  of  the  tangent  plane  to  the 
locus  of  (443)  is  <T\<f>p  +  y]  (o-  +  p)  =  C.  If  p  does  not  satisfy 
(443),  the  equation  is  that  of  the  polar  plane. 

(11)  In  Ex.  (6)  substitute  for  the  second  quadric, 

pI<^ip+2y|p  =  C, 

not  concentric  with  the  first,  and  show  that  the  locus  is  then 

p\(f>2~^<t>i^p-h2p\(l>2-^<t>iy  +  y\(f>2~\  =  {C—  pyf. 

160.  Center  of  the  general  quadric  surface.  When  the  origin 
is  at  the  center  of  a  surface,  terms  of  odd  degrees  must  not 
appear  in  the  equations,  for  it  must  be  unchanged  when  —  p 
is  put  for  +  p.  If  then  we  change  the  origin  in  eq.  (443)  by 
putting  p  +  8  for  p,  and  cause  the  first  degree  terms  to  vanish, 
the  origin  Avill  be  at  the  center.     Eq.  (443)  becomes  thus 

(p  +  8)!c^(p  +  S)+2(p  +  8);y  =  C, 

or  p\<l>p  +  2 p\<f>8-{-8\<l>8+ 2 py-\- 2 8\y=z  a 


Chap.  V.]  SOLID   GEOMETRY.  189 

If  the  first-degree  terms  are  to  vanish  for  all  values  of  p,  we 
must  have  ^8  +  y  =  0,  or 


say,  so  that    mo  =  (X/iv)  ~^<f>X<l)iJL(fiv 

and  \l/y=  (\fjt.v)'^(\(lifi<f>v  •  \\y  +  etc.) 


h  (^9) 


|.   .     .     .     (450) 


The  quantity  mo  is  an  invariant,  as  we  saw  in  Art.  155. 
The  vector  )/^y  is  also  an  invariant;  that  is,  its  length  and 
direction  are  independent  of  the  vectors  X,  fi,  v,  as  may  be 
shown  by  substituting  IX  -f  m/n  +  nv  for  any  one  of  the  three. 
It  also  appears  from  the  fact  that,  as  (443)  represents  a  defi- 
nite surface,  when  (f>  and  y  are  given,  its  center  must  be  a 
definite  fixed  point,  and  hence  8,  the  vector  to  this  point,  must 
have  a  definite  fixed  value,  so  that,  mo  being  invariant,  ijry  must 
be  so  likewise.     If  in  (449)  we  have 

mo  =  0  and  TVy^O, (451) 

the  center  is  at  co  in  the  direction  of  {jry,  and  the  surface  is  said 
to  be  non-central.  But,  by  Art.  154,  mo  =  0  is  the  condition 
that  p\(f>p  shall  be  the  product  of  two  factors  of  the  first  degree ; 
i.e.  that  it  shall  have  the  form  p|^i  •  pl/Sg-  Also,  since  the  cen- 
ter is  at  00,  take  the  origin  on  the  surface  by  making  the  con- 
stant term  disappear ;  i.e. 

8\<i>8  +  2y\8-C=0, (452) 

a,n  equation  that  gives  two  definite  values  for  T8  when  the 
direction  of  8  is  assumed.     Eq.  (443)  thus  becomes 

p\(3,-p\p,  +  2{<l>8  +  y)\p  =  0, (453) 

which  is  identical  in  form  with  (410),  and  must  therefore  rep- 
resent an  elliptic  or  hyperbolic  paraboloid. 


190  DIRECTIOXAL  CALCDX.ITS.  [Art.  101. 

If  we  have 

mo  =  0  =  Tif/y, (454) 

the  value  of  8  is  indeterminate,  and  <^8  =  —  y  is  then  the  equa- 
tion of  the  line  or  plane  of  centers,  the  locus  being  in  this  case 
either  a  cylinder  or  two  planes. 

Resuming  now  the  general  value  of  S,  and  substituting  it, 
eq.  (443)  reduces  to 


p!,^p  =  C-S|<^8-2y!8  =  C  +  y'<^-V 
=  C  +  m„-  V^i/^y  =  C",  say 


}.     .     .     (455) 


The  sign  of  C  will  be  ahvays  taken  as  positive. 
If  we  have 

C'  =  C+y\<t>-'y  =  0; 

that  is,        m^C  +  yil/y  =  0, (456) 

then  (455)  becomes 

P\<I>P  =  0, (457) 

which,  being  independent  of  Tp,  must  represent  a  cone,  real  or 
imaginary,  which,  when  8  is  indeterminate,  breaks  up  into  two 
real  or  imaginary  planes. 

16L  Maximum  and  minimum  values  of  Tp.  Tp  will  be  a 
maximum  or  minimum  when  dTp  =  0 ;  i.e.  when  p\dp  =  0. 
Eq.  (455)  gives  dp\<j>p  =  0;  hence,  at  the  maximum  or  mini- 
mum points,  p  and  (fyp  are  both  X  to  all  values  of  dp,  which 
requires  them  to  be  parallel  to  each  other ;  i.e.  we  must  have 
p<f)p  =  0,  an  equation  whose  solution  will  give  the  required 
values  of  p. 

This  equation  is  equivalent  to 

<f>P  =  (jp  or   {<j>-g)p==0, (458) 


Chap.  V.]  SOLID   GEOMETliY.  191 

g  being  a  scalar  constant  to  be  determined.     Multiply  the  com- 
plement of  (458)  by  p,  and  we  have 

p\<f>p  =  C'  =  g(^, (459) 

which  gives  the  relation  between  (Tpj^ax.  and  g. 

By  reference  to  Art.  Ill  it  will  be  seen  that  the  equation  to 
be  solved  here  is  identical  in  form  with  the  one  there  treated. 
Hence  it  will  only  be  necessary  to  put  vectors,  say  p,  \,  p,,  v, 
in  place  of  the  points  p,  q^,  q.2,  q^,  in  Arts.  Ill  and  112,  in 
order  to  obtain  results  for  a  vector  system  in  solid  space  cor- 
responding to  those  for  a  point  system  in  plane  space.  We 
will  also  substitute  g  for  n,  and  m  for  k.  Thus  we  have, 
from  (341), 

p\X4>-g)X^p\(<t>-9)f^  =  pl{<i>-g)y  =  0;  .     (4G0) 
from  (342), 

(c)>-g)\(cf>-g)p{cf>-g)v  =  0^ 

^  "  ,  a  t;  •     •     •     •     (461) 

or  g^  —  m<ff-  -f-  m^g  —  rrio  —  O  ) 

in  which  mg  has  the  value  given  in  (450), 

wii  =  (A/iv)  -\X<f>p<i>v  +  p.<f>v<l>X  +  v(l>X(i>p)  I 

m2  =  {\pv)~^(Xp.<f>v -{- pv<f>X-\-vX<f>fji)  ) 

The  m's  are,  of  course,  invariants  like  the  Jo's  of  Art.  111. 
Eq.  (461)  is  called  the  discriminating  cubic,  and  plays  an  im- 
portant part  in  the  discussion  of  eq.  (455).  Eqs.  (459),  (460), 
and  (461)  give  the  complete  solution  of  the  problem  in  maxima 
and  minima;  for  (461)  gives  three  values  of  g,  say  gi,  go,  g^, 
which,  substituted  in  (459),  give  the  lengths  of  the  maximum 
or  minimum  radii  vectores,  and  substituted  in  (460)  give  the 
directions  of  the  same.  If  we  assume  Q'l  <  g^a  <  9'3j  and'let  a,  h, 
c  be  the  corresponding  values  of  Tp,  we  have 

a?=Cgf\h'=C'gf\c^=C'g,~\      .     .     .     (463) 

Proceeding  precisely  as  in  Art.  112,  we  find,  if  pi,  p2,  pz  are 
the  values  of  p  corresponding  to  flfi,  g-i,  g^, 

Pi  =  yio-i,  P2  =  9/^'2,  ps  =  93<'-3r (464) 


192  DIRECTIONAL  CALCtTLlTS.  [Art.  162. 

in  which  aj,  og,  03  are  a  unit,  normal  system  of  vectors,  so  that, 
whatever  may  be  the  original  form  of  <f>,  it  may  always  be 
transformed  into 

<f>P  =  S'ltti  •  p.o-i  +  9^2 '  p]a-2  +  g^s  •  p!«3j      ....     (4G5) 

in  which  ai-  =.a}=.  a^  =  1,  and  ailoo  =  a^W^  =  Oajaj  =  0,  as  well 
as  tt] I <^  =  021^013  =  ogj^tti  =  0,  so  that  we  have  obtained  a  sys- 
tem of  mutually  perpendicular  conjugate  diameters.  It  also 
appears,  as  in  Art.  112,  that  the  roots  of  (461)  are  always  real. 

162.  Eq.  (461)  may  also  be  written  in  the  form 

{9-9i)(9-92){9-93)  =  0  I  . 

or  9"- {9i+92-\-9s)f+i9!£l3-\-9i9i+9i92)g-9}929s=0 )  " 

From  the  form  to  which  <f>  was  shown  in  the  last  article  to 
be  reducible,  and  from  (463),  it  appears  that  the  general  equa- 
tion (455)  can  be  written 

pKV^H'^Y^^pI«3\^i, (467) 


a  J       \  Ij  J       \  c 

the  upper  or  lower  signs  to  be  taken  according  as  the  roots  of 
(461)  are  positive  or  negative.  Comparing  this  equation  with 
(414),  it  appears  that  the  general  central  equation  represents 
one  of  the  three  surfaces  considered  in  Arts.  143  and  144. 
By  Descartes'  rule  of  signs  the  positive  roots  of  (461)  are 
equal  in  number  to  the  number  of  variations  of  sign,  and  the 
negative  roots  to  the  number  of  permanences  of  sign.  Also, 
comparing  (461)  and  (466),  we  see  that  the  coefficient  of  ^  is 
-f  1,  and  that  mo  is  -f  when  all  the  roots  are  +,  —  when  one 
root  is  — ,  +  when  two  roots  are  — ,  and  —  when  three  roots 
are  — .  Thus  taking  all  the  possible  arrangements  of  sign, 
we  have, 

When  all  roots  are  positive,        +  —  +  — 

+  -  +  + 
When  one  root  is  negative,      ^  +  —  —  + 

+  +  -  + 


Chap.  V.]  SOLID   GEOMETRY.  193 

(+  4-  -  - 
When  two  roots  are  negative,  •<  +  +  +  — 

(+ 

When  all  roots  are  negative,       +  +  +  + 

We  may  also  have  one  root  zero^  when  the  cubic  reduces  to 
a  quadratic,  and  mo  =  0. 

If  two  roots  are  zero,  the  equation  is  of  the  first  degree,  and 
wii  is  also  zero. 

These  six  cases  correspond  to  the  following  surfaces  :  — 

\8t  case.     Ellipsoid;  sphere;  point  (imaginary  cone). 

'M  case.     Hyperboloid  of  one  sheet ;  cone. 

3d  case.     Hyperboloid  of  two  sheets  ;  cone. 

4th  case.     Imaginary  ellipsoid ;  imaginary  cone. 

oth  case.  Cylinder,  elliptic  or  hyperbolic ;  two  real  or  imag- 
inary intersecting  planes  :  —  because  one  axis 
is  infinite. 

6th  case.     Parallel  planes  :  —  because  two  axes  are  infinite. 

If  two  roots  of  (461)  are  equal,  the  surface  is  one  of  revolvr 
tion,  for  which  the  condition  is 

27  vhq  — 18  momimg  +  4  rrix  —  mimi  +  4  mQini  =  0.     (467  a) 

163.   Further  consideration  of  the  case  when  Mq  =  0. 

Eesuming  the  general  equation  p\<f)p  -{-2y\p=  C,  which  was 
shown  to  represent,  in  general,  a  paraboloid,  when  m^  =  0,  it 
may  be  seen  at  once  that  this  paraboloid  passes  through  the 
intersection  of  the  central  surface  p\<}>p  =  C  with  the  plane 
p\y  =  0.  But  the  condition  Mq  =  0  causes  the  equation  p\<f>p  =  G 
to  represent  a  cylinder,  elliptic  or  hyperbolic,  according  as  wij  is 
+  or  — ,  while,  if  m^  is  also  zero,  it  represents  two  parallel 
planes.  Hence  the  plane  p|y  =  0  cuts  from  the  locus  of  (455) 
an  ellipse,  hyperbola,  or  two  parallel  right  lines,  according  as 
nfix  is  +,  — ,  or  0.  The  surface  must  therefore  be  in  the  cor- 
responding cases  an  elliptic  paraboloid,  a  hyperbolic  paraboloid, 
or  2i,  parabolic  cylinder. 


194  DIRECTIONAL  CALCULUS.  [Art.   164. 

164.  The  quantity  mi)C'.  The  sign  of  this  quantity  deter- 
mines whether  (455)  represents  a  skew  or  convex  surface,  and 
its  vanishing  shows  that  the  surface  is  developable.  We  have 
seen  in  Ai-t.  152  that  the  hyperboloid  of  one  sheet  is  skew, 
while  the  ellipsoid  and  the  other  hyperboloid  are  not.  Now 
C  is  always  to  be  taken  positive,  as  stated  in  Art.  160,  so  that 
the  sign  of  m^C',  for  the  central  surfaces,  depends  only  on  mo, 
which  we  have  seen  in  Art.  162  to  be  +  for  the  ellipsoid  and 
two-sheeted  hyperboloid,  and  —  for  the  one-sheeted  hyperbo- 
loid. Also,  in  Art.  160,  we  saw  that  when  C  is  zero,  the  cen- 
tral surface  becomes  a  cone,  a  developable  surface. 

For  the  case  when  mo  =  0,  we  have 

moC"  =  71^C  -f-  yxpy  =  y  i//y. 

Applying  this  to  eq.  (404),  to  which  form  the  general  equa- 
tion was  shown  to  be  reducible  in  this  case,  we  have 

which  is  -f  for  the  elliptic  paraboloid,  and  —  for  the  hyper- 
bolic paraboloid.  If  m  =  0,  the  surface  becomes  a  parabolic 
cylinder,  and  in  this  case  mi  =  0  also,  which  agrees  with  the 
last  article.  For  the  elliptic  and  hyperbolic  cylinders  ??ioC"=0 
because  mo  =  0,  while  C"  is  finite. 

"We  give  here  a  table  of  the  results  which  we  have  obtained 
in  the  treatment  of  the  general  equation  of  the  second  degree. 

For  the  meaning  of  the  symbols  see  as  follows :  for  mo,  eq. 
(450) ;  for  wii  and  mg,  eq.  (462) ;  for  8,  eq.  (449) ;  for  C",  eq. 
(455)  ;  for  moC,  the  present  article. 


Chap.  V.] 


SOLED   GEOMETRY, 


195 


CLASSIFICATION   OF   THE   QUADRIC. 


Name  op  Surface. 

iiiqC' 

a 

c 

»'o 

TOl 

mj 

Ellipsoid. 

+ 

Finite 

+ 

+ 

+ 

+ 

Imaginary  cone  (point). 

0 

0 

u 

(( 

11 

Hyperboloid,  1  sheet. 

- 

+ 

- 

+ 

+ 
+ 

a 

Keal  cone. 

0 

0 

" 

il 

+ 

a 

Hyperboloid,  2  sheets. 

+ 

+ 

+ 

+ 

Real  cone. 

0 

" 

0 

" 

" 

" 

Imaginary  Ellipsoid. 

- 

+ 

- 

+ 

- 

Imaginary  cone  (point). 

0 

0 

11, 

(i 

11 

c 
o 

S 
J3 

Elliptic  cylinder. 

0 

0/ 

/o 

+ 

0 

+ 

+ 

Right  line. 

0 

0/ 

0 

0 

+ 

+ 

Parallel  planes. 

0 

/o 

+ 

0 

0 

+ 

Coincident  planes. 

0 

0/ 

/o 

0 

0 

0 

+ 

Hyperbolic  cylinder. 

0 

0/ 

k 

+ 

0 

- 

+ 

Two  intersecting  planes. 

0 

0/ 

/o 

0 

0 

- 

+ 

l| 

Elliptic  paraboloid. 

+ 

CO 

CO 

0 

+ 

+ 

+ 

Parabolic  cylinder. 

0 

CO 

GO 

0 

0 

Hyperbolic  paraboloid. 

- 

00 

CO 

0 

- 

+ 

165.  Exercises.  —  (1)  Find  what  surface  is  represented  by 
the  equation  (p|ei)^  +  2p|ei -pjes  +  Syjp  =  (7,  when  ciCgy  is  not 
zero,  and  also  when  it  vanishes. 

(2)  Discuss  the  equation  pr—'n?(p\i.iy-{-2p\y=C  Avhen  n>l, 
when  n  =  1,  and  when  n<l. 


196  DIRECTIONAL  CALCULUS.  [Art.  165. 

(3)  Discuss  the  equation 

pWi '  p\^i  +  pK  •  pIA  +  p1"3  •  pi/?3  +  2p J  =  C 
in  the  following  cases : 

(a)  aj  =  5  ii  +  2  to  +  4 13,  a2  =  2  ij  —  2 12  —  3 13,  03  =  4  ij  —  3  to  — 13, 
/81  =  h,  A  =  ^2j  ^3  =  ^35  y  =  0,  C  =  1. 

(5)/33  =  ai,)8,  =  a2,  )8,  =  a3,  C=2aSy  =  0. 

(c)  The  same  as  (6)  except  y  =  —  ai  +  So.,  —  403. 

(d)  The  same  case  as  (6)  except  add  2 pr  to  the  first  member. 

(e)  ai  =  ^1  =  ai2  +  &I1,     ao  =  /^o  =  &t3  +  CI2,     ag  =  ^3  =  Cij  +  ai3, 
y  =  0,(7=1. 

(4)  Discuss  the  following  equations  : 
(a)  {e,py  +  {uj>r-  +  (e^y-  =  a\ 

(6)  cip  •  £,£3  •  £3/)  =  c^. 

(C)     (£ip  •  €2€3)e3(«2P  "  ts^l)  =  C^- 

(d)  ep^p  =C,  (f>  being  as  in  (415). 
In  the  last  case  we  may  write 


€p<f>p  =  C/J«/>p  •  l]l2*3  = 


talc,    lalp,    i2|<^pi=C 
isIc)    hip,    h\4>p\ 


(5)  Show  that,  if  p\<i>ip  =  1  and  p\<f>2p  =  1  have  parallel 
axes,  then  p|(^i  +  ^<^2)p  =  1  has  its  axes  parallel  to  theirs. 

(6)  Find  the  condition  that  the  surface  of  the  second  order 
passing  through  the  common  line  of  two  quadrics  shall  be 
developable,  and  thus  show  that  in  general  four  different  cones 
pass  through  this  common  line. 

(7)  Find  the  axes  of  a  central  plane  section  of  a  central 
quadric ;  also  the  area  of  the  curve  of  section. 

The  equation  of  the  surface  is  p\<j>p  =  1,  and  that  of  the 
plane  p|c  =  0 ;  hence,  for  a  maximum  or  minimum  value  of  Tp, 
we  have  dp\p  =  0 ;  also,  d/jjc  =  0  and  dp\(fip  =  0.     Therefore  we 


Chap.  V.]  SOLID   GEOMETRY.  197 

have  ep<fip  =  0,  a  cone  cutting  the  curve  of  section  at  its  maxi- 
mum and  minimum  points.     But  €p<j>p  =  0  is  equivalent  to 

<t)p  +  kp  +  fc'e  =  0,     whence  p\<f>p  =  l=  —  Jcp-. 
Also,    p  =  —  k'((f)-\-  k)  ~%  whence  c|p  =  0  =  c|  («^  +  k)  ~h. 
By  (441)  this  is  equivalent  to 

mo€|^"^'£  +  ke\{m2  —  <f>)€-\-  k^^  =  0. 

Area  =  Trajfti  =  — - —  =  —    ^  if  Tt  =  1. 

(8)  Perpendiculars  are  drawn  from  p  on  the  four  faces  of  a 
tetraedron,  the  feet  of  the  perpendiculars  being  coplanar ;  find 
the  locus  of  |>. 

(9)  Find  the  locus  of  a  point,  the  sum  of  the  squares  of 
whose  distances  from  n  fixed  points  is  constant. 

(10)  Find  the  locus  of  a  point,  the  sum  of  the  squares  of 
whose  distances  from  two  fixed  right  lines  is  constant. 

(11)  Find  the  locus  of  a  point,  the  sum  of  the  squares  of 
whose  distances  from  three  fixed  right  lines  is  constant. 

Determine  the  nature  of  the  locus  in  each  case. 

(12)  Find  the  condition  that  the  plane  (r|e  =  D  shall  be  tan- 
gent to  the  surface  of  eq.  (443). 

Ans.    D  +  y|<^-^£  =  ±  V6"e|<^-ic. 


Chap.  VI.]  SCALAR    POINT   EQUATIONS.  207 


CHAPTER   VI. 

SCALAR   POINT   EQUATIONS   OF  THE   SECOND  DEGREE  IN 
SOLID   SPACE. 

166.  Differentiation.  As  it  will  be  necessary  in  this  chapter 
to  use  dP,  we  proceed  to  determine  its  meaning.  Reasoning 
precisely  as  in  Art.  79,  we  see  that,  if  p  lie  on  some  surface, 
dp  must  be  a  vector  along  some  tangent  to  the  surface  at  p ; 
i.e.  a  point  at  oo  in  the  tangent  plane  at  p.  Then,  if  P=  [p, 
P  will  envelop  some  surface,  and  we  have  dP  =  \dp,  so  that 
dP  is  a  plane  through  the  mean  point  of  the  reference  tetrae- 
dron.  Also,  as  dP is  the  limit  of  P—  P'  as  these  planes  ap- 
proach coincidence,  it  must  always  pass  through  the  common 
line  of  P  and  P',  and  hence,  at  the  limit,  through  the  point  of 
contact  of  P  with  the  surface  it  envelops. 

167.  Tlie  general  homogeneous  equation.  We  shall  deal  only 
with  homogeneous  equations ;  for,  by  (377),  4:p\e  =  1,  so  that 
any  term  of  lower  degree  than  the  highest  can  be  raised  to 
that  degree  by  multiplying  it  by  the  proper  power  of  4=p\e, 
without  changing  thereby  the  meaning  of  the  equation. 

Any  homogeneous  equation  of  the  second  degree  in  p  may 
be  written  in  the  form 

P\<l>P  =  0, (468) 

in  which  <^  is  self-conjugate ;  for  such  an  equation  can  always 
be  reduced  to  the  sum  of  such  terms  as  Aip\qi  •p\qi;  that  is, 
to  the  form  %(Ap\q  -p^q')  =  0.     This  is  equivalent  to 

^lp\{Aq  'p\q'  +  Aq'  .p|g)]  =p\^[A(q  'p\q'  +  q'  ■p\q)'] 

=p\<l>p  =  0, 
if  we  write 

<t>p  =  -S,lA{q'p\q'  +  q'.2>\qn 


208  DIRECTIONAL  CALCULUS.  [Art.  168. 

168.  Eq.  (468)  represents  a  surface  of  the  second  order;  i.e. 

it  is  pierced  in  tivo  points  by  any  right  line.  The  demonstra- 
tion is  precisely  as  in  Art.  105,  and  leads  to  the  same  value 

of  ^  given  in  (327).     The  equation 

qiP\<t>qi<t>P  =  0 (469) 

now  represents  a  circumscribed  cone  with  its  vertex  at  qi,  while 

^\<f>€(l>p (470) 

represents  a  circumscribed  cylinder  ||  to  c. 

169.  Diametral  planes.  The  equation  of  the  locus  of  the 
middle  points  of  a  system  of  chords  ||  to  e  is  found,  as  in  Art. 
106,  to  be 

p\<i>^  =  0, (471) 

which  represent  now  a  diametral  pZane,  conjugate  in  direction 
to  c  Every  line  in  this  plane  is  also  conjugate  in  direction  to 
c.  If  p>  recede  to  oo,  it  becomes  a  vector,  say  e',  and  hence 
€'\<f>€  =  0  is  the  condition  for  conjugate  directions.  If  we  take 
a  second  vector  e"  ||  to  the  plane  of  (471),  then  we  have  also 
c"|^c  =  0,  and  if  we  have  also  £'|<^£"  =  0,  then  c,  c',  e"  form  a 
system  of  conjugate  directions ;  i.e.  they  are  ||  to  a  set  of 
conjugate  diameters  of  the  surface  p]<j>p  =  0. 

170.  Significance  of  the  quantity  \<^p.  Differentiating  (468) 
we  have  dp\<j>p  =  0 ;  hence  \(fip,  which,  by  (468),  is  a  plane 
through  p,  is  the  locus  of  the  tangents  to  the  surface  at  p;  i.e. 
the  tangent  plane.    Its  equation  may  be  written 

g!#  =  0 (472) 

Suppose  \<f>p  to  pass  through  some  fixed  point  e ;  then  we 
shall  have  e\<f>p  =  0  =p\(f>e.  The  point  p  in  this  equation  is  the 
point  of  contact  of  the  tangent  plane  1^^);  hence,  if  p  vary 
subject  to  the  above  condition  2)\<f>e  =  0,  this  equation  is  that 
of  the  locus  of  p,  and  is  a  plane,  the  j)olar  plane  of  e.  If  e'  be 
on  the  polar  plane  of  e,  we  have  e'|<^e  =  0  =  e]</>e',  so  that  e  is 


Chap;  VI.]  SCALAR   POINT   EQUATIONS.  209 

also  on  the  polar  plane  of  e'.  The  points  e  and  e'  are  conjugate 
to  each  other.  We  see,  by  (471),  that  a  diametral  plane  is 
simply  the  polar  plane  of  a  point  at  oo  ;  hence  the  polar  planes 
of  all  points  in  a  diametral  plane  pass  through  the  same  point 
at  00 ;  i.e.  they  are  parallel  to  the  diameter  conjugate  to  this 
plane.  If  a  point  move  along  a  line,  its  polar  plane  turns  about 
a  line.  For  let  xpi  +  yp2  be  any  point  on  the  line  pii)^  >  then 
its  polar  plane  is 

\<i>{^i  +  yih)  =  ^Mh  +  y\i>P2, 

a  plane  through  the  common  line  of  \<j>j)i  and  \<f>p2.  Also,  since 
the  polar  planes  of  points  on  \<f>2)i  and  \<f>2}.2  pass  respectively 
through  j)i  and  jXr,  it  follows  that  the  polar  planes  of  points  on 
\4*Pi^P2  pass  through  jhP-i-  If  we  put  ci  and  e^  for  jyi  and  p2,  we 
see  that  the  polar  planes  of  points  on  a  diameter  pass  through 
a  common  line  at  cc,  i.e.  they  are  parallel. 

171.  Center  of  the  surface.  This  is  the  common  point  of  any 
three  diametral  planes ;  hence,  if  e^,  cg,  £3  are  any  points  at  00, 
we  have  for  the  center 

q,  =  n\<l>€i(ji€2<t>e3, (473) 

n  being  a  scalar  factor  so  taken  as  to  make  q^  a  unit  point. 
To  evaluate  n,  multiply  both  sides  of  (473)  into  4  |e. 

.'.  4  q^le  =  1  =  4  w|^ci<^c2<^c3e  =  —  4  ne(f>€i(ji€2<f>€s. 

_        \<f>ei<f>e2<f>€s        .  ,      ^ (474) 

If  we  have 

e<f>ti<li£2<f>€3  =  0, (475) 

while  the  numerator  of  (474)  is  not  zero,  then  the  center  is  at 
00,  and  (468)  represents  a  non-central  surface.  If  we  put  q^ 
for  qi  in  (469),  we  have  the  asymptotic  cone 

pqc\<i>p4>gc  =  o (476) 

172.  Sets  of  conjugate  points.  Any  four  points  q^,  gg?  ^s,  q* 
which  satisfy  the  six  conditions, 

9ii<^5'2  =  qiHz  =  qi\Hi  =  q^M*.  =  ^4l«^9'2  =  92i«^^3  =  O,   (477) 


210  DIRECTIONAL   CALCULUS.  [Art.   173. 

form  a  set  of  conjugate  points.  Each  point  evidently  lies  on 
the  polar  plane  of  each  of  the  others ;  i.e.  the  four  points  form 
a  tetraedron  such  that  each  vertex  is  the  pole  of  the  opposite 
face.  There  is  an  infinite  number  of  such  sets  of  points ;  for, 
take  any  point  in  space  as  gj ;  then  any  point  in  |  cf>qi  as  q^ ;  by 
Art.  170,  \(J3q2  also  passes  through  q^ ;  next  take  any  point  in 
\<f>Qii>Q2  ^^  Qsi  then  j^gg  passes  through  gj^j  by  Art.  170,  and 
cuts  \<f>qi(f>q2  in  g^. 

If  q^  be  at  co,  gig^gs  is  a  diametral  plane ;  if  q.^  be  also  at  oo, 
then  qiq2  must  be  on  a  diameter,  and,  finally,  if  g^  be  at  oo, 
gi  must  be  the  center  of  the  surface ;  thus  g^,  e,  e',  e"  are  a  set 
of  conjugate  points,  three  of  which,  being  at  oo,  reduce  to 
directions,  as  in  Art.  169. 

173.  Normal  set  of  conjugate  2Mints.  If  four  points,  besides 
the  conditions  (477),  satisfy  also  the  following 

Qi\Q2  =  qi\qs  =  qi\qi  =  q3\Qi  =  9*192  =  92193  =  0,   .    .    (478) 

they  may  be  called  a  normal  set  of  conjugate  points.  We  pro- 
ceed to  show  that,  with  reference  to  any  given  surface  J)\^P=0, 
there  is  one  and  only  one  normal  system  of  conjugate  points . 

174.  Solution  of  the  equation  cjyp  =  np.  Write  the  equation 
(<^  —  n)p  =  0,  and  multiply  its  complement  successively  by 
any  four  points  e,  e',  e",  e'",  thus  obtaining 

e  \{^  —  n)p=p\{^—n)e  =0 
e'  \{<^  —  n)p=X)\{^<^  —  n)e'  =0 
e"\{4>-n)pz=p\{<i>-n)e"  =0 
e"'|(</)  -  n)p  =2)\(<f>  -  n)e'"  =  0 

Each  of  these  equations  must  be  satisfied  by  the  same  values 
of  p  that  satisfy  the  given  equation ;  therefore  p  must  be 
simultaneously  on  the  four  planes  |(<)!)  —  n)e,  etc. :  hence  these 
four  planes  must  have  a  common  point,  the  condition  for 
which  is 

(<^  -  n)e(cf>  -  n)e'{4>  -  n)e"{4>  -  n)e'"  =  0, 


(479) 


i      -     (480) 

or         n*  —  kn7i^  -\-  k^n-  —  k^n  +  k„  =  0,  ' 


}, (^81) 


Chap.  VI.]  SCALAR    POINT   EQUATIONS.  211 

in  which  the  ^-'s  have  the  following  values  : 

ko  =  {ee'e"e"')  -'cf>e<fie'<l>e"<i>e"' 
k,  =  (ee'e"e"')  -'•S,(e<l>e'<i>e"<f>e"') 
ko  =  {ee'e"e"')-^%{ee'<f>e"<f>e"') 
ks  =  (ee'e"e"')-':^{ee'e"<l>e"') 

the  summations  being  on  the  following  plan,  viz. :  the  </)'s  are 
to  be  placed  before  the  different  e's  in  succession,  three  by 
three,  two  by  two,  and  one  by  one,  for  ki,  k-r,  and  k^,  respec- 
tively ;  thus  ee'(f>e"ct>e"'  +  cf>ee'e"cf>e"'  +  etc. 

The  solution  of  (480)  will  give  four  values  of  n,  which,  sub- 
stituted in  (479),  will  give  the  required  points.  Eq.  (480) 
can  have  no  imaginary  root.  This  may  be  shown  precisely  as 
in  Art.  112. 

Let  the  roots  be  Uj,  n^,  rig,  n^,  to  which  correspond  the  points 
Pi>  Pn}  Ps)  Pi ;  then  these  points  are  given  by  the  equations 

Pi\{<f>  —  ni)e((j>  —  ni)e'{<f>  —  ni)e"  =  0  ^ 

p,\{<f>  -  7i,,)e(<^  -  n2)e'(<^  -  H2)e"  =  0 

etc.,     etc. 

in  accordance  with  (479). 


(482) 


175.  The  points  pi,  •  •  •  ^4,  just  determined,  constitute  a  normal 
set  of  conjugate  points.  Since  <f>2h  =  niPi,  <f>p2  =  ihPij  ©tc,  we 
have  Pi\<i>po,  =  thP\\p-2,  etc. ;  so  that,  '\ipi\P'z  =  0,  then  Pi\<l>P2  =  0 
also.  As  <^  is  self -conjugate,  write  it  in  the  most  general  form 
of  such  a  function  in  terms  of  Wj,  •••  W4  and pj,  '--Pi;  i.e. 

2<t>p  =  -  niPi  •  ihPsPiP  +  «2i^2  •  PsPiPiP  -  W3P3  •  PdhPiP 

+  ^iPi  -IhPaPsP  —  '>h\p2PzPi  'Pi\p  +  n<i\PsPiPi  -pip 
—  '>h\PiPiP2-Pz\P  +  ni\pi2hPs  'Pi\P- 
In  order  that  this  form  of  ^  may  satisfy  the  above  condi- 
tions, we  must  have  PiP-2PiPi='^,  Pr  =  Pr  =  Pi  =  Pr  =  ^j  and 
Pi\P2  =  Pi\Pz=Pi\Pi=Pz\PA—pi'P2=Pi'P3  =  ^-    These  last  con- 
ditions also  imply  \p2P3Pi  =  —Pi,  etc.,  for  they  make  pi  lie 


212  DIRECTIONAL   CALCULUS.  [Art.  176. 

simultaneously   on   the   three  planes   \p2,   \2hy   \Pi  S    therefore 
j9i  =  in\p2Psi\,  m  being  a  scalar  constant.     Multiply  into  \pi. 

•  ••  2h-  =  1  =  mlPiPsPiih  =^  -  mhPiPiPi  =  —m, 
so  that         2h=  —  \P-22hPi- 

Thus,  whatever  the  original  form  of  <f>  may  have  been,  by 
expressing  it  in  terms  of  jJj,  •••2h  as  determined  in  Art.  174, 
it  will  always  be  reduced  to  the  form 

<PP  =  niPi ' p\px  +  n^po  •  p\P2  +  ns2h  ■p'jh  +  n^2h  •  P\P4-    (483) 

176.  Canonical  form  of  p\^p.     With  </>  as  in  (483),  we  have 

p\<f>p=')h{p\2hy+noXp\p-?)-+n^{p\Pzf+ni{p\Piy-=0,    (484) 

as  a  form  to  which  p|^p  =  0  may  always  be  reduced. 
This  equation  may  also  be  written 

pKPiVwi  +i)2V—  nj)  -i^Ki^iVni  — i^aV—  iu 

+l>|(i>3Vw3  +i^4V-  n^)  ■2^\{P!i^ns—2W-  n,)  =  0, 

or        p\qi-p\gi  +p\q2'P\q-2=^, (485) 

in  which 

qi  =i3iV%  +P2V—  n.2,  g/  =PiVni  — i>.V—  n^, 
92  =P3Vn^  +  2h'V-  Vi,  qJ  =lh^'>h  —Ih^—  ^h- 
In  this  form  it  appears  that  the  surface  passes  through  the 
four  common  lines  of  two  pairs  of  planes  \qi,  [g/  and  \q2,  l^a'* 
If  these  planes  are  all  real,  then  real  right  lines  lie  wholly  on 
the  surface ;  if  any  of  them  are  imaginary,  then  there  are  no 
real  right  lines  on  the  surface. 

177.  Rectilinear  generators.  Using  q^,  qi,  etc.,  as  in  the  last 
article,  the  pairs  of  planes 

rp\  (gi  +  mg2 )  =  0  ^  ^pj  (^i  +  nq^')  =  0  \ 

}^iw-i,/)=of-''jpiw-;',,o=oi'  '*^''> 

when  different  values  are  assigned  to  m  and  n,  each  intersect 
in  a  system  of  right  lines  on  the  surface  of  eq.  (484).     Thus 


CiiAi'.  VI.]  SCALAR   POINT   EQUATIONS.  213 

the  surface  has  two  systems  of  rectilinear  generators  when  the 
^'s  are  all  real. 

Any  two  belonging  to  the  systems  m  and  n  respectively 
intersect,  for 

(^i+mga)  (gi'  -  -  q^')  {qi+nq.J)  (g/  -  -g,)  =qiq^q2q^{l  -1)  =  0. 

//6  It 

No  two  of  the  same  system  intersect,  for 

(gi  +  m^q.^  (  gi' qA  (gi  +  m^q^)  (  g/ g^' ) 

which  is  not  zero  unless  nii  =  m^,  when  the   two   generators 
coincide. 

178.  The  discriminant.  The  expression  ^q  as  given  in 
(481)  is  the  discriminant  of  the  quantity  p\<^p,  and  is  the 
criterion  for  distinguishing  whether  (468)  represents  a  skew, 
developable,  or  convex  surface.  A^o  is  an  invariant,  as  are  also 
Tci,  Jc2,  and  kg  of  (481)  ;  i.e.  they  are  unchanged  if  any  other 
points  be  substituted  for  e,  e',  etc.  Now  we  have  seen  that  any 
form  of  (ji  may  be  transformed  by  changing  the  reference 
points  into  the  form  (483)  ;  hence,  as  this  transformation  will 
not  affect  k(„  it,  Jc^,  must  have  the  same  meaning  for  this  form 
as  for  the  original  one. 

Eq.  (483)  gives 

»  JCq  =  711712^3^^4 (487) 

Considering  positive  and  negative  values  of  Jcq  as  dependent 
on  the  signs  of  the  n's,  we  have  four  cases : 

1st.  All  the  w's  positive,  Jcq  positive  ; 
2d.    One  of  them  zero,  Zc^  =  0 ; 
3d.    One  of  them  negative,  kg  negative ; 
4th.  Two  of  them  negative,  Jcq  positive. 

Three  n's  negative  does  not  give  a  new  case ;  for,  by  chang- 
ing all  the  signs,  we  have  only  one  negative. 

In  the  first  case  all  the  g's  of  (485)  are  imaginary,  and  no 


214  DIRECTIONAL   CALCULUS.  [Art.  170. 

real  value  of  p  will  satisfy  (484),  which  therefore  represents 
an  imaginary  surface. 

In  the  second  case,  suppose  714= 0;  then  (485)  takes  the  form 

which  represents  a  cone  touching  \qi  and  |^i'  along  their  inter- 
sections with  \ps,  imaginary  if  w„  n^,  n^  are  all  positive,  real 
if  one  of  them  is  negative ;  thus  we  have  a  developable  surface. 

In  the  third  case  suppose  W4  negative ;  then  q.,  and  qj  are 
real,  while  (/i  and  q^  are  still  imaginary  ;  hence  there  are  no 
real  right  lines  on  the  surface,  which  is  therefore  convex. 

In  the  fourth  case  let  n^  and  W4  be  negative.  [It  manifestly 
makes  no  difference  which  two  are  supposed  negative,  for  any 
other  pair  as  well  as  ?i2  and  114  might  just  as  well  have  had 
negative  signs  under  the  radicals  in  (485).]  We  now  have  all 
the  5's  real,  so  that  the  surface  has  two  real  systems  of  recti- 
linear generators,  and  is  therefore  skew. 

If  rig  =  ?i4  =  0,  the  surface  becomes  simply  two  planes,  real 
or  imaginary,  intersecting  in  a  real  right  line. 

"We  have  thus 

(  positive  for  a  skew  surface  \ 

fc„(=  (fieo<f)ei<f>e2<f>es)  <  zero         for  a  developable  surface  [- .  (488) 
(  negative  for  a  convex  surface  ) 

In  the  value  of  A*o  we  have  put  Cq,  •••  e^  instead  of  e,  •••  e'", 
because  the  value  of  Tcq  is  unchanged  thereby,  and  the  reference 
points  are  the  ones  generally  most  convenient. 

179.  Nature  of  the  surface  at  infinity.  To  ascertain  this,  let 
the  variable  pointy?  recede  to  cc,  by  substituting  for  it 

Poo  =  P  =  ^'1  +  2/fa  +  2e3, (-489) 

in  the  equation  of  the  surface  p\^p  =  0. 

.-.   (x£i  +  yt^  -f  zcg)  \cf>{xti  +  y--2  +  2:3)  =  0, 
or  a^ciI«Afi  +  /«2l<^e2  +  2%;  </.£3  +  2  2/2;c2|«^e 


+  2zx^\4,ii  +  2  xy€i\<j>€i 


=0    }•  ■  •  "*> 


Chap.  VI.] 


SCALAR   POINT   EQUATIONS. 


215 


Eqs.  (489)  and  (490)  taken  together  represent  a  cone  pass- 
ing through  the  intersection  of  i>/^i?  =  0  with  the  plane  at  oo. 
If  this  cone  be  real,  the  surface  has  a  real  curve  at  qo  ;  if  it  be 
imaginary,  the  surface  has  no  real  curve  at  oo ;  if  it  break  up 
into  tAvo  real  or  imaginary  planes,  the  surface  has  two  real  or 
imaginary  right  lines  at  go. 

Take  ai,  oo,  a^  as  in  Art.  140,  so  that  ai  =  ItgCs,  etc.,  this  com- 
plement being  regarded  as  referring  to  a  vector  system. 

Also  Avrite  A,  B,  C,  D,  E,  F  for  the  coefficients  in  (490) 
taken  in  order,  i.e.  tj  <^£i  =  A,  etc.     From  (489), 

po-i  —  ^,   p.o->  =  y,   p\a.3  =  z, 
so  that  (490)  becomes 

A(p\a,y-  +  B(p[a,y  +  C(p\a,y  +  2  Dp\a,  •  p\a, 
+  2  Ep\as  •  pai  -\-  2Fp\ai  •  p\a^  =  0 
and        (f>'p=  (Aai+Fao  +  Eos)  •  pjaj-f-  (JFai-f  ^-f  Das)  •  Pia2 
+  {Eai-{-Da,+Cas)-p\a^. 

We  will  next  ascertain  for  this  equation  the  values  of  m^ 
»»!,  m2  appearing  in  the  table  in  Art.  164.     These  will  be  suffi- 
cient to  determine  the  cone  completely. 
We  have,  if  we  put  A.  =  cj  =  gj  —  e^,  etc., 
Vlo  =  ^'£i</>'c2«^'£3=  {Aai+Fa2+Eas)  {Fai+ Ba2-{- Dtts) 
(Ea^  +  Z>a2  +  Cog) 


]■• 


(401) 


A  F  E\ 

F  B  d\  = 
EDO 


ei!«^Ci    C2<^Ci    es'e^Ci 

Ciji^ej      <^2^f2      C3|*^«2 
Cl|«/»C3     tai^Cs     C3l<^C3 
=  (ei  —  Co)  (^2  —  ^o)  («'3  —  ^o)  '(f>^l<f>^2<t>^3 

=  4  |e  •   \(f>Ci<f}€2<f>^  =  4  e<^Ci^£2*^^3 
«ll  =  £i<^'£2^'£3  +  £20'£3^'£i  +  f^s'f*' ^I't*' ^2 


=^  «lC2«sl<^«l^«2«A^3 


.(492) 


\B  D\ 


+ 


C  E\ 
E  a\ 


+ 


A  F\ 
F  b\ 


=  ^2^3\4>^2<i>^3  +  £3Ci:<^C3(^£i  +  €i£2l<^£j</»£2  ^ 
Vlo  =  £i£2<^'£3  +  ^2^s<f>'^l  +  ^3^l4>'^i  =  C  +  A-\-  B 
=  «li^«l  +  «2  </>f2  +  ^s\*i>^3 


(493) 


(49-1) 


216 


yDIEECTIONAL  CALCULUS. 


[Art.  180. 


Note  that  wio  is  the  denominator  of  q^  in  Art.  171,  where  we 
have  already  determined  the  meaning  of  mo  =  0.  By  the  aid 
of  the  values  just  found  for  the  m's,  A'o  of  (488),  and  q^  of 
(474),  together  with  the  table  in  Art.  164,  Ave  construct  the 
following  table  for  the  quadric  in  a  point  system. 


Name  op  Sckfacb. 

<?c 

io 

J»0 

m, 

m, 

Ellipsoid. 

Finite. 

- 

+ 

+ 

+ 

Point  (imaginary  cone). 

" 

0 

t( 

(( 

(( 

Imaginary  ellipsoid. 

i( 

+ 

ii 

(( 

(( 

Elliptic  paraboloid. 

tX) 

- 

0 

+ 

+ 

Elliptic  cylinder. 

% 

0 

0 

+ 

+ 

Parabolic  cylinder. 

% 

0 

0 

0 

+ 

Hyperbolic  cylinder. 

% 

0 

0 

- 

+ 

Hyperbolic  paraboloid. 

00 

+ 

0 

- 

+ 

Hyperboloid  of  two  sheets. 

Finite. 

- 

+ 

+ 

+ 
+ 

Cone. 

n 

0 

(( 

(( 

(1 

Hyperboloid  of  one  sheet. 

(( 

+ 

(( 

(( 

(( 

180.  In  (484)  put  ^o?  ••• -^g  for   nj,  •••?i4   and   Cq, -"ea  for 
Pi,•"P^,  ^o  that  the  equation  becomes 


tiA-{p\ey]=0, 


(495) 


and  (483)  becomes  2[u4e  -pie]  =  <^2^.  Then  we  have  <^eo=-4oeo) 
^fii  =  A-fii,  etc. ;  ^ci  =  ^(e^  —  e^)  =  A^Bi  —  A^fi^,  etc.  Hence  we 
find 


Chap.  VI.]  SCALAR   POINT   EQUATIONS.  217 

^'       A>-'  +  A,-'  +  Ar'+A,-'      ' 

"'0  =  AqA-i^zAs, 

mo  =  A1A2A3  +  A^AsA^  +  AsA^i  +  A0A1A2, 
Ml  =  2  Ao{Ai  +  ^2  +  ^3)  +  A^As  +  ^3^1  +  A1A2, 
m2  =  3Ao-\-Ai  +  A2-\-As. 

181.  Consider  next  the  equation 

P\eo(Ap\ej,  +  Bp\e2  +  Cp\e^).-\- p\ei{C'p\e2  +  -B'^leg) 

+  ^'^162.^163  =  0, (496) 

which  represents  a  quadric  surface  passing  through  the  four 
reference  points,  since  it  is  satisfied  when  p  =  e,,,  p  =  gj,  etc. 
It  is  the  most  general  equation  of  the  second  degree  in  p 
representing  a  locus  passing  through  these  points ;  for  it  con- 
tains all  the  combinations  of  the  quantities  pjeo,  pl^i,  etc., 
taken  two  at  a  time,  and  there  must  be  no  term  containing 
only  one  of  these  quantities,  as  p\e^^,  because  this  term  would 
not  vanish  when  p  =  e^. 

Eq.  (496)  contains  five  arbitrary  constants,  and  can  there- 
fore be  made  to  fulfil  five  conditions,  such  as  to  pass  through 
five  given  points;  but  the  locus  already  passes  through  the 
four  reference  points ;  hence  the  general  quadric  can  be  sub- 
jected to  nine  conditions,  and  the  general  equation  must  con- 
tain nine  arbitrary  constants.  We  shall,  in  fact,  obtain  this 
general  equation  by  adding  together  (495)  and  (496) . 

182.  Conditions  in  order  that  (496)  shall  represent  a  sphere. 
If  (496)  represent  a  sphere,  the  sections  of  the  surface  by  the 
reference  planes  must  be  circles.  Take  the  section  by  the 
plane  p\eQ  =pexe^s  =  0,  and  the  equation  becomes 

A'p\e.i  -2)163  +  B'p\e^  •p\ei+C'p\ei  'p\e^  =  0. 

Consider  the  plane  space  fixed  by  ej,  e^,  63;  then,  by  (358), 
the  condition  that  this  equation  shall   represent   a  circle   is 


218 


DIRECTIONAL   CALCULUS. 


[Art.  183. 


A^      B'      C 

—^  =  —5=^,   in   which    025=^6263,   asi=  Te^e^,    a,o.=  Te^e^ 

a<Q  0,31  Clyj 

Proceeding  in  the  same  way  with  the  other  reference  planes, 
we  find  the  required  conditions  to  be 


«01 


B^ 

0.02' 


^03 


aJ 


an 


(497) 


183.  Exercises.  —  (1)  Show  that  for  eq.  (496)  we  have  the 
following  values,  in  which  2^  =  B -\-C -  A',2^  =  G +  A-B', 
26  =  ^  +  jB-C',  viz.: 


aSq  — 


Wo=  — ' 


0  A  B   C 

AQ   C  B' 

1 

B  C  0  A' 

'   ^'  =  ^ 

C  B'  A'  0 

Cq,  61,  €.2,  63 

A,  -A,  C'-B,  B'-C 

B,  C'-A,       -B,  A'-C 
C,B'-A,A'-B,       -C 


^6  33 

33  sr  C 


AQ 
6  B 


+ 


15  2t 


+ 


cm 


m,' 


-2(A  +  B-\-C). 


(2)  Show  that  for  the  equation  formed  by  adding  (495)  and 
(496),  we  have 


ko  — 


Wo  = 


A,A  B   0 

A  A,  C  B' 

B   C'A.A' 

,      fJc  = 

C  B' A' As 

€(),  €1,  B21  ^3 

A  -  Ao,  A,  -A,C'-B,B'-C 
B  -  Ao,  C  -A,Ao-B,A'-C 
C -  Ao,  B'  -  A,  A'  - B,  As-  C 


W  6"  33" 
6"  33'  21" 
3]"  21"  6' 


,  mi  = 


2t'  6" 
6"  33' 


+ 


33'  21" 
21"  6' 


+ 


6'  33" 
33"  21' 


+  6', 


m2=3l'  +  : 
in  which 

2l'  =  A  +  A-2^,  ^'  =  Ao  +  A,-2B, 
Q,'  =  A>  +  A.,-2C,  %"  =  A,-\-A'-B-C, 
^"  =  A,-\-B'-C-A,  (S"  =  A,+  C'-A-B. 


Chap.  VI.]  SCALAR   POINT   EQITATIOKS.  219 

(3)  Show  that  (496),  with  the  conditions  (497),  represents 
a  sphere,  by  transforming  the  equation  to  a  vector   system. 

See  Art.  75. 

(4)  Show  that  the  most  general  equation  of  the  second 
degree  in  p,  in  solid  space,  may  be  written 

p\eo-p\eo  +P\ei-p\ei  +2^h-p\e2'+Ph-ph'  =  0;    .     (498) 

write  the  corresponding  self-conjugate  form  of  <f>,  and  deter- 
mine <^eoj  "-(fx^s- 

(5)  Show  that  planes,  tangent  to  (49G)  at  the  reference 
points,  cut  the  opposite  reference  planes  in  four  right  lines, 
which  are  generators  of  the  same  system  of  a  skew  quadric. 

(G)   Show  that  the  four  lines  of  Ex,  (5)  are  coplanar  if 
AA'  =  BB'  =  CC. 

(7)  Prove  the  proposition  —  of  which  Ex.  (5)  is  a  particular 
case  —  that  the  corresponding  faces  of  any  tetraedron  and  its 
polar  tetraedron  with  reference  to  any  quadric,  cut  each  other 
in  four  right  lines  which  are  generators  of  the  same  system  of 
a  skew  quadric. 

(8)  Show  that  the  conditions  given  in  Ex,  (G)  make  eq. 
(496)  represent  a  convex  surface. 

(9)  Find  the  condition  that  the  plane  p\e  =  C  shall  be  tan- 
gent to  the  quadric  2^\^P  =  0,  determining  thus  the  equation 
of  the  tangent  plane  independent  of  the  point  of  contact.  By 
using  the  result,  find  the  tangent  plane  conjugate  in  direction 
to  c. 

Ans.   4  Ce\(j>~'^e=e\^~^e  ±  V— eel^~^e<^~^e,  4p[<^e  =  ±  i  ~^' ,  • 

(10)  If  a  plane  be  tangent  to  any  quadric  p\<l>2P  =  0,  show 
that  the  locus  of  its  pole,  with  reference  to  any  other  quadric 
p\cf>ip^i),  is  q\(t>.i^<lii'q  =  0. 


220  DmECtlOJfAL  CALCULUS.  [Art.  181. 

(11)  Find  the  nature  of  the  surfaxje  in  each  of  the  following 
cases,  the  position  of  the  center,  and  the  asymptotic  cone,  if 
real : 

(a)  In  (496)  let  A  =  B  =  0=  A'  =  B'  =  C  =  1.    Ellipsoid, 

(b)  In  (496)  let  C"  =  0.     Skew  or  developable  surface. 

(c)  In  (496)  let  B'=C'  =  0.     Skew  surface. 

(d)  In  (496)  letA  =  B=C=0.     Cone,  vertex  at  e^. 

(12)  In  cases  (&),  (c),  (d)  of  the  last  exercise,  show  that 
one,  two,  and  three  edges  respectively  of  the  reference  tetrae- 
dron  lie  on  the  surface. 

(13)  Show  how  the  cone  of  Ex.  (11),  case  (d),  is  related  to 
the  surface 

(14)  Discuss,  as  in  Ex.  (11),  the  following  cases : 

(a)  In  (496)  let  ^  =  J5  =  C  =  1,  A'  =  B'=C'  =  2.  Ellip- 
soid. 

(b)  In  (496)  let  A  =  B=C  =  A' =  B'  =  1,C' =3.  Elliptic 
paraboloid. 

(c)  In  (496)  let^  =  -i?  =  e=-^'=5'=-C"=l.  Two- 
sheeted  hyperboloid. 

(d)  In  (496)  let  A  =  B=C  =  A' ==B'  =  1,C'  =  0.  Elliptic 
cylinder. 

(e)  In  (496)  let  A=  -  B=  C=  -  A' =  B',  C  =  0.  Cone, 
vertex  at  ^{62  -\-  e^). 

(/)  In  (495)  let  A^  =  A^  =  Ai= -As  =  \.  Two-sheeted 
hyperboloid. 

(gf)  In  (495)  let  A^  =  Ai=  —  A^^  —  A^.  Hyperbolic  para- 
boloid. 

{h)  In  (495)  let  A  =  2,  ^1  =  ^2  =  ^,  A.,=  -  1.  Elliptic 
paraboloid. 


Chap.  VI.]  SCALAR    POINT   EQUATIONS.  221 

(i)  In  (498)  let  e^' =  i{e,-\-eo-\-e^),  ei' =  i{e,  +  e.,  +  e^), 
^  =  -JCes  +  ^0  +  ei),  e/  =  ^(^.1  +  ^i  +  e.). 

(i)  In  (498)  let  e^  =  Ci  +  e,  +  63-260,  e/  =  eo  +  ^3  +  ?o  -  2e„ 
c/  =  e.-!  +  ^'o  +  ei  —  2^2,  63'  =  e,,  +  ^'i  +  62  —  2^3. 

(A)  In  (498)  let  e^  =  «/  =  e,  e.,'  =  62,  ei  =  63. 

(0  In    (498)    let    e^^e^    ej'  =  |(e„  +  e,),    e/  =  i(Po  +  e2), 

(15)  Discuss  the  equation  XfJ^-Jj-JL^j)  =  0,  and  show  that  it  is 
the  locus  of  a  right  line  moving  on  three  rectilinear  directrices. 

(16)  A  tetraedron  has  one  variable  vertex  p,  and  three  fixed 
vertices  Cj,  Cj,  63;  the  variable  faces  pcifia,  etc.,  pass  through 
the  points  in  which  ig,  L^,  Lo  pierce  a  variable  plane  which 
always  passes  through  e^:  show  that  the  equation  of  the 
locus  is 

{Li  -pe^z)  {L2  -pe^i)  (X3  -^6162)  60  =  0. 

(17)  Three  planes  pass  respectively  through  the  lines  Zj, 
Lt,  Ls,  and  two  of  the  common  lines  of  these  planes  alwaj'S 
cut  Li  and  L2'  respectively ;  the  locus  of  the  third  common 
line  is  a  quadric  whose  equation  is  pL^Li  L^L^  L.yp  =  0. 

(18)  There  are  given  four  lines  L^,  e^t^  L.>,  e^^ ;  two  planes 
pass  respectively  through  L^  and  L^,  cutting  e^cx  and  €.^2  in  two 
points  which  move  along  these  lines  at  rates  bearing  a  con- 
stant ratio  to  each  other ;  show  that  the  locus  of  the  common 
line  of  these  planes  is  a  quadric  passing  through  L^  and  L^ 
whose  equation  is  Z  •  j)eiZi  •  ^^2-^2  =  ^'i  •  P^ 2-^2  •  peiA-  Consider 
the  cases  when  I  and  m  are  of  like,  and  unlike,  signs,  and 
when  L^L,  =  0. 

(19)  Show  that  the  two  surfaces  p\4>p  =  0  and  p\<^p  =  G  are 
similar,  concentric,  and  similarly  placed. 

(20)  Discuss  the  following  equations  : 

{pe^y  =  <?,    {pe^Y  ±  {pe,y-  =  c^, 
{pe,y-  +  {pe,Y±{pe.;)'-  =  (?, 


222  DIRECTIONAL  CALCULUS.  [Art.  184. 

I^  =  c-,    (i>eoei)==(r, 

{pe^i)-  ±  {pe<fis)-  =  (r, 

{pe^e^Y-  ±  (peoe2)-±{iyeoesY  =  c^. 

(21)  Find  the  equation  of  the  surface  reciprocal  to  2)[<^p=0; 
i.e.  the  locus  of  g  =  ^j9. 

184.  Inversion  of  <f>.  If  we  have  given  cf)2)  =  q,  whence  we 
have  J)  =  <f>~^q,  we  must  be  able  to  invert  <f>  in  order  to  find  p 
as  an  explicit  function  of  q.  Taking,  for  generality,  cf>  as  not 
self-conjugate,  let  <^c  be  the  conjugate  function,  so  that 

q\<i>p=p\<f>cq- 

Let  e,  e',  e",  e'"  be  any  four  points ;  then  we  have 

e\(f>p  =p\<f>,e  =  e\q,  e'l^p  =p\^,e'  =  e'|g, 

e"|(^|?  =  P\^fi"  =  e"'^,         e"'\<f>p  =  p^,e'"  =  e'"\q. 
Now  substitute  in  (201)   c^^e  for  jpo»  ^c^'  for  p„  etc. 

If  we  put  q  =  \e'e"e"',  which  we  can  do  whatever  point  q 
may  be,  because  the  e's  are  any  points  whatever,  and  hence 
three  of  them  may  always  be  so  taken  that  the  complement  of 
their  product  shall  be  q,  then  (499)  reduces  to 

Jcocli-ye"e'"  =  \<l>,e'4>,e"<l>,e"', (500) 

ko  having  the  value  given  in  (481)  except  that  <^<,  must  be  put 
for  <f>. 

Now  substitute  <^  4-  n  for  <t>,  where  n  is  any  scalar. 

.-.  Jco'{<t>  +  ny'q  =  ko'{cf>  +  n)-'\{e'e"e"') 

=  i  (<^,  +  n)e'{<f>,  +  n)e"{<f>,  +  n)e"'. 

On  expanding  the  last  member,  the  first  term  is  ^'o*^"  ^q,  the 
last  is  n^q,  and,  as  the  first  member  is  a  function  of  q,  it  ap- 
pears that  the  coefficients  of  n  and  ?r  should  also  be  functions 


Chap.  VI.]  SCALAR   POINT   EQUATIONS.  223 

of  q.     Call  them  xq  and  \pq  respectively ;  then,  expanding  Tcq\ 
the  equation  becomes 

{n^+kzn^+lcjfi^+Tiin+TcQ) {<f>-\-n)-^q  =  Jc.^  \-\-nxq+n-xj,q+n\. 

Operate  on  this  equation  by  <^  +  n. 
.'.   (n*  +  Jc^n^  +  k.,n^  +  JciU  +  k^i)q 

=  kf,q-{-  Jc^n  4>-^q+ n<l>xq + n'xq + n^tfi^q  -\-  n^if/q + n^<t,q + n*q. 

This  equation  must  hold  for  all  values  of  ?i ;  hence  the  co- 
efficients of  the  same  powers  of  n  on  each  side  of  the  sign  of 
equality  must  be  equal. 

.•.  ij/  =  A'o  —  <^,  and  X  =  ^'-2  —  ^':)4>  +  •^■• 

Also,     k,,<i>-\  =  kiq-'k.^q-\-hi4rq-<i/'q,    ) 

...     (oOl) 
or  {<t,'-ks<f>'  +  k2<l>'-ki<t>-{-k,)q  =  0.\  ^       ^ 

Substituting  the  values  of  xf/  and  x  ^^  the  value  of  (<;^  +  n)~', 
we  obtain  also 

/j^  ■  j^y\f       h<l>~^q-hn(k2-ks<i>  +  i>^)q+7i\h-cf,)q-^n^q     .^^. 
^  n*  +  ksiv^ -[- k^n- +  kiu  +  kf^  ^ 

185k  Equation  of  the  anti-polar  plane.  This  is  found  pre- 
cisely as  in  Art.  121,  and  is  of  the  same  form,  viz. : 

pl<^(2ry,-e)  =  0, (503) 

e  being  the  point  of  wliich  this  equation  represents  the  anti- 
polar  plane. 

186.  Reciprocating  ellipsoid.  To  obtain  the  equation  of  this 
surface  we  proceed  in  the  same  manner  as  in  Art.  122  for  the 
reciprocating  ellipse. 

Assmne  the  equation 

p\ii>p  =p\e^ .  p!(ei  +  e.,  +  e^)  +p\e^  -p\{e.  +  e,)  +p|e2  -ple^ 
_±mi4.p\ey  =  0, 

and  determine  m,  so  that,  with  reference  to  this  surfiute,  each 


224  DIRECTIONAL  CALCULTTS.  [Art.  187. 

reference  point  shall  be  the  anti-pole  of  the  opposite  reference 
plane.     The  equation  gives 

^i>  =  (8 me  +  ei  +  62  +  63)  . p|eo  +  (8 me  +  62  +  63  +  eo)-p\ei 
+  {8me  + 63  +  60  +  61)  'p\62  +  {8me +60  +  61  +  63) .i)|e3. 

The  center  of  the  surface  is  at  e,  so  that  (503)  becomes 
p\<f>{2e  —  e)  =  0,  or,  putting  60  for  e,  and  its  value  for  e, 
p\(j){6i  +  e2  +  e3  —  eo)  =  0.  We  must  then  have  [co  coincident 
with     1^(61  +  ^2  +  63  —  60),     whence     60^(61  +  62  +  63— 60)  =  0. 

Now 

60^(61  +  62  +  63  —  60)  =  60  [8  me  +  62  +  63  +  eo  +  etc.] 
=  60  [16  me  +61  +  62  +  63] 
=  (4m  +  l)eo(ei  +  6.,  +  63). 

This  expression  becomes  zero  if  m=  —^,  which  value,  on 
substitution,  gives  for  the  required  equation 

p\eo'P\iei  +  e,  +  e,)+p\ei.2:>\{6.2  +  e3)  )         . 

)- .       .      (504) 
+P\e2-P\ez-A(,p\ey  =  0^  ^       ^ 

Also,  (f>p  may  be  written 

</>i>  =  (61  +  62  +  63  -  60)  •  p\6o  I 

+  (62  +  63  +  60  —  61)  -plei  +  etc.  )  ' 

187.  We  will  now  show  that  the  complement  of  any  point  is 
its  anti-polar  plane  with  reference  to  the  ellipsoid  of  (504.)  If 
this  is  true,  we  must  have 

\p-\cl,{2e-p)  =  0  ox  2y<f>{2e-p)  =  0. 
Let  p  =  2oW6,  with  the  condition  ^pi  =  1 ;  then 
p<f>(2e—p)  =  2we  •  <^[(i -  rio)eo  +  (i  -  Wi)ei  +  etc.] 
=  2ne  •  [  (i  -  Wo)  (61  +  e,  +  e.  -  e^)  +  etc.] 
=  2w6  •  [(1  —  «i  —  n.,  —  ??3  +  n^,)e^y  +  etc.] 
=  2p2ne  =  2pp  =  0.     q.k.d. 


Chap.  VI.]  SCALAR   POINT  EQUATIONS.  225 

188.   Scalar  plane  equations.     In  eq.  (496)  put  Pfor  \p,  and 
we  obtain  the  complementary  or  anti-polar  reciprocal  equation 

Pe„ .  P{Ae,  +  Be,  +  Ce,)  +  Pe,  •  Pi(J'e,  +  B%)  )  ,  ^^^^ 

+  ^'Pe,.Pe3  =  0  ['  •     ^"''^ 

which  is  that  of  a  surface  touching  the  four  reference  planes, 
since  it  is  satisfied  when  P  =  |e(„  P=  |ej,  etc.     If  we  write 

\^P={Ae^+Be,+Ce^)  •  Pe„+  {C'e.+B'e^+Aeo)  •  Pe, 

+  {A'e,+Beo+  Ce,)  •  Pe,+  {Ce,-^B'e,-}-A'e,)  •  Pe^ 
eq.  (506)  becomes 

P|./rP=0, (508) 


X,  (507) 


and  i/f  is  a  linear,  self-conjugate  function  of  P. 

In  the  same  way  any  scalar  point  equation  p\<jip  =  0  may  be 
changed  into  its  complementary  plane  equation  P|i/'P=0. 
Bearing  in  mind  that,  if  P=  \p,  then  |P=|(|p)  =  —p,  we  have 
the  following  relations  between  <^  and  if/,  viz. : 

ij/P  =  ip\p  =  \<l>p  =  -\ct>\P. (509) 

Suppose  (508)  to  represent  any  homogeneous,  plane  equa- 
tion of  the  second  degree.  The  equation  shows  at  once  that 
the  point  \i(/P  lies  on  the  plane  P.     Differentiating,  we  have 

dPli/^P-f  P|./.dP=  2dP|>/.P=  0; 

but,  by  Art.  166,  dP  is  a  plane  through  the  point  of  contact  of 
P  with  the  surface.  Hence  [i/^P  is  on  the  line  Pf?P.  Now  dP 
is  a  variable  plane,  subject  only  to  the  conditions  of  always 
passing  through  e  and  the  point  of  contact  of  P;  hence  IfP 
must  coincide  with  the  point  of  contact  of  P.  If  Q  be  any 
plane  not  tangent  to  (508),  then  [ij/Q  is  its  pole  with  reference 
to  that  surface. 

189.  Center  of  surface  P|i/^P=0.  The  center  is  the  pole 
of  the  plane  at  oo,  i.e.  of  |e;  hence,  by  (509), 

Qc  =  m\f\e  =  m\  (|^e)  =  —  m^e. 


226  DIRECTIONAL   CALCULUS.  [Art.  I'M. 

To  evaluate  m,  multiply  into  4  |e. 

.-.  4  g„|e  =  1  =  —  m(f>e  •  4  |e  =  —  4  me|^e ; 

whence  o.  =  .  I* ,  _ (510) 

If  we  have  e\4>e  =  0, (511) 

the  center  is  at  cc,  and  the  surface  is  a  paraboloid.  This  is 
also  the  condition  that  the  reciprocal  surface  |)|^j>  =  0  shall 
pass  through  the  mean  point  of  the  reference  tetraedron. 

190.  Reciprocal  surfaces.  A  skew  surface  is  reciprocal  to  a 
skew  surface.  For  such  a  surface  is  generated  by  a  right  line 
whose  consecutive  positions  are  not  coplanar;  hence  the  recip- 
rocal surface  is  generated  by  a  right  line  whose  consecutive 
positions  have  no  common  point. 

A  developable  surface  is  reciprocal  to  a  curve,  and  vice  versd. 
For  such  a  surface  may  be  regarded  as  the  envelope  of  a  plane 
rolling  on  some  two  given  surfaces  Si  and  So ;  hence  the  point 
reciprocal  to  this  plane  lies  simultaneously  on  the  two  sur- 
faces reciprocal  to  Si  and  S2;  i.e.  it  generates  their  common 
line.  When  the  developable  surface  is  a  cone,  the  reciprocal 
curve  \&  pilane;  for,  since  all  tangent  planes  to  the  cone  have  a 
common  point,  their  reciprocal  points  are  coplanar. 

A  convex  surface  is  reciprocal  to  a  convex  surface.  This  fol- 
lows from  the  preceding,  since  all  surfaces  are  included  under 
these  three  heads. 

19L  The  discriminant  k^  as  a  criterion.  Since,  by  (509), 
i{/\p=\<l>p,  we  have,  by  the  last  article,  and  by  (488),  that  when 

(  +  )  C  skew  surface     ) 

k^){=  i{/,eQ{l/\ei\p\e.^\e._i)  is  -^  0  y,P\il/P  =  0  is  a  4  plane  curve       /-. 
(  —  )  (  convex  surface  ) 

192.  To  determine  still  farther  the  surface  represented  by 
(508)  we  will  make  use  of  the  cone  circumscribed  about  the 
surface  p|(^p  =  0  and  having  its  vertex  at  e,  the  center  of  re- 
ciprocation.    If  this  cone  is  real,  the  surface  cuts  the  plane  at 


Chap.  VI.] 


SCALAR    POINT   EQUATIONS. 


227 


CO  in  a  real  curve,  the  reciprocal  of  this  cone.  If  the  cone  is 
imaginary,  the  surface  has  no  real  points  at  x. 

Substituting  e  for  q^  in  (469),  we  have  the  equation  of  the 
above-mentioned  cone,  viz. : 

'pe\^p<^e  =  i) (512) 

Write  (fi'iy  =  cjij)  •  e\(f)e  —  (fie  •  2)\4>e, (^l^^) 

and  with  this  value  of  (^'  compute  m^,  m^,  vu  according  to  Art. 
179,  and  determine  by  the  table  in  the  same  article  whether 
(.512)  is  real  or  imaginary.  AVe  obtain  thus  the  following 
scheme  for  the  determination  of  the  equation  Pif/P=0. 


Surface. 

/Cq 

Cone  of  eq.  (512). 

Ellipsoid. 

- 

Imaginary. 

Ellipse. 

0 

(( 

Imaginary  ellipsoid. 

+ 

" 

Elliptic  paraboloid. 

- 

Two  coincident 
planes  tangent  at 
e.    {e\(i-e  =  0). 

Parabola. 

0 

Hyperbolic  paraboloid. 

+ 

Hyperboloid  of  two  sheets. 

- 

Real. 

Hyperbola. 

0 

(( 

Hyperboloid  of  one  sheet. 

+ 

u 

193.  Exercise. — If  (^  and  i//,  p  aii'^^  J\  are  related  as  in 
Art.  188,  show  that  2>|<^  ^p  =  0  and  P\^P=  0  are  respectively 
the  point  and  plane  equations  of  the  same  surface,  reciprocal 
to  that  represented  by  p\^p  =  ^.  Also  that  P|i/^'^P=0  and 
p\<^p  =  0  are  respectively  the  plane  and  point  equations  of  a 
surface  reciprocal  to  that  represented  by  P\\pP=  0. 


Chap.  VIL]  APPLICATIONS   TO   STATICS.  237 


CHAPTER   VII. 

APPLICATIONS   TO   STATICS. 

194.  It  is  proposed,  in  this  concluding  chapter,  to  give  a 
few  applications  of  our  calculus  to  mechanics,  merely  to  serve 
as  an  introduction  to  this  field,  and  to  indicate  how  perfectly 
the  methods  that  have  been  developed  adapt  themselves  to 
mechanical  conceptions  and  processes. 

195.  A  force  is  that  which  is  postulated  as  the  cause  of  any 
change,  or  tendency  to  change,  in  the  rate  of  motion  of  some 
particle,  or  rigid  body,  on  which  it  acts. 

In  statics  we  consider  systems  of  forces  aside  from  any  ques- 
tion of  rest  or  motion  of  the  bodies  on  which  they  act ;  and, 
especially,  cases  in  which  the  total  resultant  effect  of  all  the 
forces  applied  to  a  body  is  null. 

196.  The  space  qualities  of  a  force  are  magnitude,  direction 
and  2)osition,  and  these  are  the  only  qualities  with  which  we 
have  to  do  mathematically.  The  intrinsic  character  of  a  force, 
such  as  that  of  gravity  or  magnetism,  we  know  little  or  nothing 
about;  but  our  knowledge  is  complete  for  its  mathematical 
treatment,  when  we  know  its  magnitude,  direction,  and  line  of 
action,  or  position.  Xow  these  space  qualities  are  identical 
with  those  of  sl  point-vector ;  hence,  for  the  purposes  of  mathe- 
matical discussion,  a  point-vector  completely  represents  a  force, 
and  therefore  all  that  has  been  demonstrated  in  Chapter  II. 
regarding  the  former  can  be  applied  immediately  to  the  latter. 

197.  Notation.  The  notation  for  points  and  vectors,  in  gen- 
eral, will  be  as  in  previous  chapters.  The  vector  representing 
the  magnitude  and  direction  of  a  force  will  be  denoted  by  a 


238  DIRECTIONAL   CALCULUS.  [Art.  198, 

German  letter  as  g^,  while  the  magnitude  of  the  same  force 
will  be  denoted  by  F,  the  corresponding  English  letter.  Thus 
i^  will  be  the  tensor  of  '^,  or  F=  T^-.  If  e  be  a  point  on  the 
line  of  action  of  the  force,  it  will  then  be  completely  denoted 
by  e"^,  a  notation  which  is  practically  more  convenient  than 
the  use  of  a  single  letter  to  represent  the  point-vector  or  force. 
The  complement  will  be  used  in  this  chapter  only  with  refer- 
ence to  a  vector  system  in  solid  space. 

198.  Forces  acting  on  a  ^xirticle.  The  parallelogram  and 
polygon  of  forces  follow  at  once  from  the  nature  and  properties 
of  vectors  and  point-vectors,  as  shown  in  Chapters  I.  and  II. 
Let  a  system  of  forces  acting  on  the  point  e  be  denoted  by 
e^i,  6^25  •"  ^^^n  j  then  the  resultant  effect  of  the  system  will  be 
found  in  this,  as  in  all  cases,  by  simply  adding  the  forces.     Thus 

Resultant  =  SeJ  =  eSJ (514) 

For  equilibrium  we  must  have  e%'^  =  0,  or 

25  =  0, (515) 

an  equivalent  equation. 

199.  Equilibrium  of  a  2)article  contained  to  remain  on  a  smooth 
curve.  In  this  case  the  resultant  force  must  have  no  compo- 
nent along  the  tangent  to  the  curve,  at  the  point  where  the 
particle  is ;  hence  the  resultant  must  be  A.  to  this  tangent. 

Let  the  equation  of  the  curve  be 

p  —  eo  =  p  =  (f>t, 
<f>  being  a  vector  function  of  the  scalar  variable  t ;  then 

dt      dt 

is  a  vector  along  the  tangent.  Hence,  if  j9  be  the  position  of 
the  particle  on  the  curve,  the  condition  for  equilibrium  is 

||25  =  ||25  =  </>'«|2S  =  0 (516) 


Chap.  YU.]  APPLICATIONS   TO   STATICS.  239 

For  example,  if  the  curve  become  the  right  line  whose  equar 
tion  is  p  =  €  +  e't,  then   dp  =  ddt,  and  the  condition  becomes 

Again  consider  the  case  of  a  particle  resting  on  a  diagonal 
of  a  parallelopiped,  and  acted  on  by  three  forces  represented  by 
the  edges  of  the  parallelopiped  which  meet  at  a  corner  not  on 
the  diagonal.  Let  the  three  edges  be  tj,  ej,  e^,  and  the  equation 
of  the  diagonal 

thus  25  =  ei  +  £2  +  C3,  and  ^  =  e^  +  £3  -  ^i, 

so  that  we  have 

(^2  +  Cg  —  Ci)  I  (ci  +  £2  +  C3)   =  0, 

or  (e2  +  £3)i_e,2  =  0;  i.e.   Te,=  T{t,  +  ^,). 

200.  Equilibrium,  of  particle  constrained  to  remain  on  a 
smooth  surface.  If  v  be  a  vector  ||  to  the  normal  at  p,  then  for 
equilibrium  2^  must  be  ||  to  this  vector. 

.-.  v25  =  0 (517) 

is  the  required  condition.  If  the  equation  of  the  surface  is  a 
scalar  one  in  terms  of  vectors,  then  it  will  be  linear  in  dp  after 
differentiation  and  will  have  the  form  v\dp  =  0,  v  being  some 
function  of  p.     To  illustrate,  let  the  equation  of  a  surface  be 

(phy-{-(p\e,r-ph  =  0; 

then  3  (p[ei)^  •  dp\€i  +  2p[£2  •  dple^  —  dp\€s  =  0, 

or  dpi (3 £1 .  (p\e,y  +  2  £2 .  pl£2  -  £3)  =  dp'v  =  0. 

If  we  have  a  vector  equation,  it  will  be  in  the  form 

p  =  <f>(x,y). 

Then  -^  and  -B.  will  be  vectors  ||  to  tangents  to  the  surface  at 
dx         dy 

the  end  of  p,  and  hence  we  may  write 

^-P'Ip (518) 

dxdy 


240  DIRECTIONAL  CALCULUS.  [Art.  20L 

201.  Examples. —  (1)  If  e^,  eo,  e^  are  the  vertices  of  a  tri- 
angle, and  2h}  Ihf  Ps  the  middle  points  of  its  sides,  pi  opposite 
Ci,  etc. ;  then  forces  represented  by  SiPi,  e^po,  e^Ps  are  in  equi- 
librium. 

"We  have 

2e5  =  e,p,  -f  e,p,  +  e,p,  =  e/^^A  +  e,(^-^\  -f-  ^3^^^^) 
=  i  (6163  +  e^es  +  6,^3  +  62^1  +  e^ei  -f-  e.e.^  =  0. 

(2)  Forces  are  represented  by  perpendiculars  dra^vn  from 
the  vertices  of  a  triangle  on  the  opposite  sides ;  to  show  that 
they  cannot  be  in  equilibrium  unless  the  triangle  is  equilateral. 

Let  the  triangle  be  616263 ;  a  =  Tlegea,  b  =  Te^e^,  c  =  Te-fi^ ;  2h 
foot  of  _L  on  62^3;  etc. ;  I,  m,  n  cosines  of  angles  at  Cj,  e.,,  e^. 
Then, 

Pj  =  - (6/162  +  cmeg),    2h  =  -  (ci^3  +  ttnci) ,    p^  =  -  {amci -\- blei) , 
a  be 

and 

^e%=eiPi-\-e2P2+esPs=nf — t)^i^^-+\1 — )e2«3+w(---Wi> 

which  cannot  be  zero  unless  a  =  b  =  c. 

(3)  Let  £1,  cj,  €3  be  any  three  unit  vectors,  and  eciF^,  ee^F^, 
ecgFg  three  forces  acting  at  e.     Then  for  equilibrium 

2g  =  %^F  =  CiFi  +  62^2  +  C3F3  =  0. 

Tliis  gives  cie2C3  =  0,  so  that  the  forces  must  be  coplanar.  Also, 
ci«2^2  =  csci^s)  and  £iC2-Fi  =  t^^F^ ;  i.e.  t^toFf^  =  cgCgF,-^  =  ^f^iFf^. 
But  the  three  vectors,  being  coplanar,  may  be  taken  as  in  plane 

space,  and,  therefore,  c,£,,  etc.,  scalar ;   so  that  qe,  =  sin  <  ^-, 

etc.,  and  we  have 

Fi  :-sin  <  ^^  ^  F2 :  sin  <  '^  =  2^3 :  sin  <  ^'-^ 

^2  ^3  ^1 

(4)  If  £1,  £2,  £3,  £4  be  any  four  unit  vectors,  and  ee^Fi,  et-^F.^,  etc., 
be  four  forces ;  show  that  for  equilibrium  we  have 

^sU^f^  =    —  £3£4eii^2~^  =  U^l^i^Z^  =    —  ^l^i^S^i'^- 


Chap.  VII.]  APPLICATIONS   TO   STATICS.  241 

(5)  Show  tliat  the  point  on  the  smooth  surface 

where  a  particle  attracted  toward  the  origin  will  remain  at 
rest,  is  given  by 


a^       W       <f       -^a^^b^  +  c^ 

(6)  Through  a  point  at  the  end  of  e  three  chords  are  drawn, 
parallel  to  a  set  of  conjugate  diameters  of  a  central  quadric ; 
forces  act  on  this  point,  represented  by  the  portions  of  the 
chords  intercepted  between  the  point  and  the  surface,  and 
towards  the  surface  :  show  that  2^  =  —  2e. 

Let  a,  )8,  y  be  conjugate  semi-diameters,  and  equations  of 
chords  p  =  e-\-xa,  p  =  e-t-  yfi,  p  =  € -{- zy.  Substitute  the  first 
value  of  p  in  the  equation  of  the  surface ; 

.-.    (€-\-Xa)\<f>(€  +  Xa)=l  =  c\<l)e-\-2xa\<f>€  +  a:^, 

because  a|<^a  =  1.  Now  the  sum  of  two  of  the  forces  will  be 
i^-i  +  '^2  =  {^i-\-^2)"-}  ill  which  Xi  and  ajg  3-^6  the  roots  of  the 
preceding  equation. 

•'•  ?5^i  +  3^2=  —  2a«al<^€. 
Hence  we  have 

Sg:  =  -  2(a  .  t\^a  +  P  '  t<f>(3 -\- y  €cf>y) 


=  -2(^ 


c/?y  +  /3  •  cya  +  y  -ea^>,  -  _  2  < 


aySy 
The  last  results  are  by  eqs.  (423)  and  (177). 

(7)  Show  that,  if  a  system  of  forces  acting  on  a  point  are 
represented  by  vectors  drawn  outward  from  the  point,  and  are 
in  equilibrium  ;  then  this  point  is  the  mean  of  the  extremities 
of  the  vectors. 


242  DIRECTIONAL   CALCULUS.  [Art.  202 

202.  Forces  acting  on  a  rigid  body.  Let  the  forces  be  Ci^i, 
e^i,  etc. ;  then  we  have  for  the  total  resultant  effect 

2B  =  Seg-  =  e,%%  +  2  (e  -  eo)  ^  =  e,%'^  +  Sc^.       (519) 

In  the  last  member  we  have  written  ci  for  ej  —  e^,  etc. 
Comparing  with  Art,  61,  we  see  that  the  quantity  2S  has  the 
same  space  qualities  as  the  quantity  there  denoted,  by  S,  and 
called  a  screw.  Following  again  the  nomenclature  of  K..  S. 
Ball,  Astronomer  Eoyal  of  Ireland,  we  shall  call  the  quantity 
2B  a  wrench,  and  it  is  evident  that  it  corresponds  to  a  screw, 
just  as  a  force  does  to  a  point-vector. 

Before  proceeding  to  a  general  discussion  of  eq.  (519),  we 
will  consider  some  special  cases. 

203.  Parallel  forces.  Let  3^i=i^ie,  %2=F^,  etc.,  and  T(.=l; 
then 

2B  =  2e^  =  %eF€  =  -  aeF=  -  te^F,    .     .     (520) 

in  which  e  is  the  mean  of  all  the  points  e^,  Cg,  etc. 

Now  e  depends  only  on  gj,  e^,  etc.,  and  will  be  the  same  what- 
ever the  direction  of  e  may  be ;  hence  it  is  a  unique  jjoint,  with 
reference  to  the  system  of  ||  forces,  possessing  the  property 
that  through  it  the  resultant  always  passes,  no  matter  what 
may  be  the  direction  of  the  forces.  It  is  called  the  center  of 
II  forces,  and  is  determined  by  the  equation 

e  =  2Fe  H-  2F. (521) 

In  this  case  the  wrench  SS  reduces  to  ix  force  acting  at  e. 

204.  Couples.     Suppose  that  we  have 

2^  =  0; (522) 

then  (519)  becomes 

2S  =  2e^, (523) 

so  that  3B  reduces  to  a  ^Zane-vector.     Consider,  first,  one  plane- 
vector  only,  of  those  that  make  up  ^,  for  instance 

ciS^i  =  (ei  -  Co)  5i  =  Cj^i  -  eQ%x ; 


Chap.  VII.]  APPLICATIONS   TO   STATICS,  243 

it  appears  as  the  sum  of  two  unlike  parallel  forces,  of  equal 
magnitude.  This  is  denominated  a  couple,  and  its  only  effect 
on  the  body  is  to  produce,  or  tend  to  produce,  rotation, 
cj^i  =  Tei^i '  Uei'^i  and  the  first  factor  evidently  measures  the 
magnitude  of  the  rotational  effect ;  i.e.  the  product  of  Fi  into 
the  X  distance  between  gj^i  and  ei,^'y„  which  is  called  the 
moment  of  the  couple.  All  the  properties  of  couples  follow  at 
once  from  those  of  plane-vectors,  as  demonstrated  in  Chapter 
II.  Eq.  (523)  gives  the  resultant  of  all  the  couples  acting  on 
the  body,  and  therefore  the  total  tendency  to  produce  rotation. 

205.  Condition  for  a  single  resultant  force.  By  eq.  (218) 
the  condition  that  2S  shall  reduce  to  a  single  force  is 

2B^  =  0  =  (^0^5  +  2c5)^  =  2  eoS^Se^. 

Hence  the  condition  may  be  written  either 

22^  =  0,  or  2^2e^  =  0 (524) 

This  shows  that  the  plane  of  the  resultant  couple  must  be 
parallel  to  the  resultant  force. 

Eq.  (524)  may  also  be  satisfied  by  Se^^  =  0,  which,  by  (519), 
reduces  25^  to  a  single  force ;  and,  by  SJ  =  0,  which  reduces 
the  wrench  to  a  couple,  or  zero  force  at  oo.  It  follows  that  any 
system  of  forces  confined  to  one  plane  will  be  equivalent  either 
to  a  single  force,  or  to  a  couple. 

206.  For  equilibrium  we  must  have 

2B  =  eo25  +  2e^  =  0, (525) 

which,  because  2B  is  the  sum  of  a  point-vector  and  a  plane- 
vector,  requires  that  we  have 


%%  =0 


=  0} (^^^> 


207.  Normal  form  of  a  wrench.  It  appears,  by  Art.  67,  that, 
by  properly  choosing  the  point  at  which  the  resultant  force 
acts,  5B  may  always  be  reduced  to  the  sum  of  a  force  at  q,  and 
a  couple  whose  plane  is  perpendicular  to  the  force. 


244  DIRECTIONAL  CALCULUS.  [Akt.  208. 

Writing  in  (215)  S^r  for  a,  and  2eg  for  \/3,  we  have 

SS  =  eoS3^  +  2e3^  =  ^2^+^||f.|%      •     •     .     (527) 

(2g)- 

and,  by  (214),  the  vector  perpendicular  between  epSJ  and  q'S% 
is 

(m'       (2g^^' ^'^^^ 

in  which  p  =  q  —  €(,. 

By  Art.  46,  the  second  term  of  the  third  member  of  (527) 
is  the  orthogonal  projection  of  the  resultant  couple  2eJ  on  a 
X)lane  ±  to  2§,  i.e.  on  12^.  Kow  the  orthogonal  projection  of 
any  plane-vector  upon  a  plane  is  always  less  than  the  projected 
plane-vector  in  magnitude;  hence  the  couple  in  the  normal 
form  of  2B  is  the  minimum  of  all  the  couples  obtained,  when 
^0  occupies  positions  not  in  the  line  of  q%%  which  is  called  the 
axis  of  the  wrench. 

206.  Recurring  to  eq.  (213),  let  us  consider  the  equation 

T{\p-pa.)=C, (529) 

i.e.  the  area  of  plane-vector  part  of  S  constant. 
Taking  the  co-square,  we  have 

^--2paP  +  {pay-  =  /3'--2pafi+fM-  {p\ay  =  C, 

or         (U-{p\ay--2paP=C'-l3^ (530) 

Writing  (f>p  =  a-p  —  a'  p\a,  so  that  the  first  two  terms  be- 
come p\<f)p,  and  comparing  with  (443),  we  have 

y  =  -  \al3,  and  (C  of  (443) )  =  C^  -  /^.     .     .     .     (531) 

In  (449)  put 

X  =  a,  fi  =  /3,  v=  |a/?, 

SO  that 

<fya  =  0,   <t>l3  =  a^(3  -  a  ■  ^\a,   <f>\a{3  =  J  -  \af3, 

and  we  see  that  8  —  -,  so  that  the  surface  represented  by  (529) 


Chap.  VII.]  APPLICATIONS   TO   STATICS.  245 

is  a  cylinder.  We  have,  as  in  Art.  IGO,  for  the  equation  of 
the  axis  of  this  cylinder  <f>8  +  y  -0] 

i.e.   a-8  —  a-  S\a  —  \a/3  =  0, 

or,  by  (189),      a8,a=ia^, (532) 

which  is  identical  with  (214),  from  which  (528)  was  derived. 
TJms  the  axis  of  the  cylinder  coincides  with  the  axis  of  the  screw, 
qa. 

The  discriminating  cubic  (461)  becomes  {g  —  a-Y  =  0,  so 
that  the  cylinder  is  circular.  Now  putting,  as  in  the  last 
Article,  2^  for  a  and  Seg^  for  \/3,  we  see  that  the  locios  of  q, 
when  the  moment  of  the  resultant  couple  is  constant,  is  a  circular 
cylinder  whose  axis  is  the  axis  of  the  wrench. 

209.  Any  lorench  can  he  reduced  in  an  infinite  number  of 
icays  to  the  sum  of  two  forces.  Let  us  write  2^J  =  TF(ee  +  aje), 
to  which  form  we  have  seen  that  any  wrench  is  reducible. 
Now  put  Wa\f.  =  {e' —  e)^,  which  requires  e' —  e  and  ^  to  be 
both  J_  to  e,  but  can  be  satisiied  by  an  infinite  number  of 
values  of  e'  and  '^,  subject  to  these  conditions.  Substituting, 
we  have 

22  =  We,  +  (e'  -  OS"  =  <  We  -  %)  +  e% 

which,  by  what  we  have  just  seen,  proves  the  proposition. 

The  tetraedron  of  which  the  opposite  edges  are  any  two 
forces  whose  sum  is  9S  is  of  constant  volume.  Let  p^i  and 
j)^2  be  two  such  forces,  so  that  p^i  +i>2^2  =  2B.  Therefore, 
squaring, 

2p,%,p.^,  =  ^'  =  2e,%^%,%. 

The  first  member  of  this  equation  is  12  times  the  volume  of 
the  before-mentioned  tetraedron,  and  the  last  member  is  a  con- 
stant when  2B  is  given,  which  proves  the  proposition. 

210.  Exercises.  —  (1)  Find  the  conditions  in  order  that 
the  orthogonal  projections  on  any  direction  whatever  of  a 
system  of  forces  may  be  themselves  a  system  of  forces  in 
equilibrium. 


246  DIRECTIONAL   CALCULUS.  [Akt.  210. 

Let  I  be  a  unit  vector  in  any  direction ;  then  the  projection 
of  ^1  on  I  is  I  •  ij^i,  and  similarly  for  the  other  forces.  By 
(526)  we  have  for  equilibrium,  putting  i  •  t|?yi  for  ^i,  etc., 

2(.-.J5)  =  t.i|:SJ  =  0. 

As  this  is  to  be  true  for  all  directions  of  i  we  must  have 
%^  =  0.  Again,  writing  i-tj^  for  ^  in  the  second  of  (526), 
we  have 

2(«.t|g)=-.2(£.c:5)  =  0 

as  the  other  condition,  which  requires  2(£  •  l'%)  to  be  ||  to  i. 

(2)  Forces  act  at  the  vertices  of  a  tetraedron,  in  directions 
respectively  J.  to  the  opposite  faces,  and  proportional  to  the 
areas  of  these  faces  in  magnitude ;  show  that  the  forces  have 
the  property  discussed  in  Ex.  (1). 

Let  CoCij  eoCj,  CqCo  be  three  edges  of  the  tetraedron ;  then, 

3^0  =  |(«1  —  fs)  (fl  —  ^2),     dl  =  |e2C3J     %2  =  k3«U     ds  =  1«1«2- 

Hence  we  have  at  once  2^  =  0.     Also, 

so  that  the  second  condition  is  fulfilled. 

(3)  A  cube  is  acted  on  by  four  forces ;  one  is  in  a  diagonal, 
and  the  others  in  edges,  no  two  of  which  are  coplanar,  and 
which  do  not  meet  the  diagonal ;  find  the  condition  that  they 
may  be  equivalent  to  a  single  force. 

Ans.  If  ^i,  ^2>  %3  a-i'e  along  the  edges,  and  ^4  along  the  diag- 
onal, the  condition  is 

(4)  Six  equal  forces  act  along  six  successive  edges  of  a  cube 
which  do  not  meet  a  given  diagonal ;  find  the  resultant  wrench. 

Ans.  If  F  be  the  magnitude  of  each  force,  and  the  edges  of 
the  cube  be  e^ii,  eoi^,  ^0^3?  we  have 
m  =  -2F\{c,  +  L,  +  cs)- 

(5)  If  503  =  ^1^1  +  €2%,  and  6162  is  -L  to  g,  and  gaj  find  the 
normal  form  of  2B  and  the  position  of  its  axis. 


Chap.  VII.]  APPLICATIONS   TO    STATICS.  247 

Let  60  =  ^(61  +  60),  and  t  =  e^  —  e,,  ='  —  (ci  —  e„).     Then 
^\%\  =  ^1^2  =  0  and  :Sc3-  =  £^2  -  £gi  =  ^(^2  -  %)■ 
Hence 

2B  =  g(^,  +  g,)  +  7|^:^- 1(^1  +  ^2). 
vyi  T"  02;- 

By  (528)  the  projection  of  g  —  e^  on  a  plane  ±  to  2^^  is 

{%.  +  %?-     ' 
so  that,  if  we  take  g  —  gy  ±  to  2^,  we  have 

(6)  Three  forces  whose  magnitudes  are  1,  2,  and  3  act  along 
three  successive  edges  of  a  unit  cube  which  are  not  coplanar ; 
show  that  the  vector  equation  of  the  axis  of  the  wrench  is 

P  =  tI'i  +  i'2  —  t\'3  +  ^('i  +  2t2  +  313). 

(7)  Forces  act  at  the  mean  points  of  the  faces  of  a  tetrae- 
dron,  J_  and  proportional  to  the  faces  on  which  they  act ;  show 
that  they  are  in  equilibrium. 

(8)  Let  p  be  any  point  within  a  tetraedron,  and  let  a  system 
of  II  forces  act  at  the  vertices,  each  proportional  in  magnitude 
to  the  tetraedron  formed  by  joining  p  with  the  other  three 
vertices ;  find  the  centre  of  ||  forces. 

(9)  The  sides  of  a  rigid  plane  polygon  are  acted  on  by 
forces  _L  to  the  sides  and  proportional  to  them  in  magnitude, 
all  the  forces  acting  in  the  plane  of  the  polygon,  and  being 
directed  inwards ;  also  the  sides  taken  in  the  same  order  are 
severally  divided  by  the  points  of  application  in  the  constant 
ratio  of  m  to  w ;  show  that  the  system  of  forces  is  equivalent 

to  a  couple  whose  moment  is  ^ — — — 1 2e-,  in  which  fx  is  the 

2(m  +  n) 

ratio  of  the  magnitude  of  any  force  to  the  length  of  the  side 

it  acts  upon. 


MATHEMATICS.  81 

The  Method  of  Least  Squares. 

With  Numerical  Examples  of  its  Application.  By  George  C.  Com- 
STOCK,  Professor  of  Astronomy  iu  the  Univei-sity  of  Wisconsin,  and 
Director  of  the  Washburn  Observatory.  «vo.  Cloth,  viii  +  U8  pages. 
Mailing  Price,  §1.05 ;  Introduction  Price,  S'l.OO. 

rpHIS  work  contains  a  preseutatiou  of  the  methods  of  treating 
observed  numerical  data  which  are  in  use  among  astronomers, 
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and  presupposes  only  such  mathematical  attainments  as  are  usually 
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Elements  of  the  Differential  Calculus. 

With  numerous  Examples  and  Applications.  Designed  for  Use  as  a  Col- 
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82 


MATHEMATICS. 


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Elements  of  the  Integral  Calculus. 

Second  Edition,  revised  and  enlarged.  By  W.  E.  Byerly,  Professor  of 
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work,  and  have  frequently  recom- 
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Its  value  is  greatly  increased  by  the 
additions.  It  is  a  fine  introduction 
to  the  topics  on  which  it  treats.  It 
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treatises  of  Todhunter  and  William- 
son, as  one  of  the  best  of  hand- 
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and  in  some  cases  leaving  out  of 
sight,  the  old  ruts  long  since  worn 
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MATHEMATICS. 


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